Fundamentals of Fluid Mechanics Bruce R. Munson Donald F

Fundamentals of Fluid Mechanics Bruce R. Munson Donald F
Fundamentals of Fluid Mechanics Bruce R. Munson Donald F. Young heodore H. Okiishi

Table Of Contents Introduction. 3 Fluid Statics. 36
Elementary Fluid Dynamics—The Bernoulli Equation Fluid Kinematics Finite Control Volume Analysis Differential Analysis of Fluid Flow Similitude, Dimensional Analysis, and Modeling Viscous Flow in Pipes Flow Over Immersed Bodies Open–Channel Flow Compressible Flow.

Chapter 1: INTRODUCTION

1.1. Historic Background Until the turn of the century, there were two main disciplines studying fluids: • Hydraulics - engineers utilizing empirical formulas from experiments for practical applications. • Mathematics - Scientists utilizing analytical methods to solve simple problems (Aero/Hydrodynamics)

1.1. Historic Background Prandtl ( ) Fluid Mechanics is the modern science developed mainly by Prandtl and von Karman to study fluid motion by matching experimental data with theoretical models. Thus, combining Aero/Hydrodynamics with Hydraulics. Indeed, modern research facilities employ mathematicians, physicists, engineers and technicians, who working in teams to bring together both view points: experiment and theory. Von Karman ( )

1.1. Historic Background Some examples of fluid flow phenomena:-
Aerodynamics design : the engagement of a wing from static state using a suitable angle of attack will produce a start vortex. The strength of it is very important for the airplane to obtain high upwards lift force, especially in aircraft takeoff on carrier. This photo shows a model wing suddenly starts its motion in a wind tunnel. Waves motion : Much of the propulsive force of a ship is wasted on the wave action around it. The distinctive wave patterns around a ships is the source of this wave drag. The study of these waves, therefore, is of practical importance for the efficient design of ship.

1.1. Historic Background Hydraulic Jump Structure-Fluid interaction
A circular hydraulic jump in the kitchen sink. Hydraulic jump is a fluid phenomenon important to fluid engineers. This is one type of supercritical flow, which is a rapid change of flow depth due to the difference in strength of inertial and gravitational forces Structure-Fluid interaction Vortices generated due to motion in fluid is of great important in structural design. The relation of a structure’s natural frequency with the shedding spectrum affect many fields of engineering, e.g. building of bridges and piers. Photo shows the vortex resembling the wake after a teaspoon handle when stirring a cup of tea.

1.1. Historic Background Tidal Bore Droplets dynamics
Tidal bore is a kind of hydraulic jump, and can be regarded as a kind of shockwave in fluid. The knowledge of its propagation is critical in some river engineering projects and ship scheduling. The photo shows the famous tidal bore in Qiantang River, China. Droplets dynamics Fluid dynamics sometimes is useful in microelectronic applications. Droplets dynamics is crucial to the bubblejet printing and active cooling technology. Photo shows a drop of water just hitting a rigid surface, recorded by high speed photography.

1.2. Fundamental Concepts The Continuum Assumption
Thermodynamical Properties Physical Properties Force & Acceleration (Newton’s Law) Viscosity Equation of State Surface Tension Vapour Pressure

1.2.1. The Continuum Assumption
Fluids are composed of many finite-size molecules with finite distance between them. These molecules are in constant random motion and collisions This motion is described by statistical mechanics (Kinetic Theory) This approach is acceptable, for the time being, in almost all practical flows

Within the continuum assumption there are no molecules. The fluid is continuous. Fluid properties as density, velocity etc. are continuous and differentiable in space & time. A fluid particle is a volume large enough to contain a sufficient number of molecules of the fluid to give an average value for any property that is continuous in space, independent of the number of molecules.

Characteristic scales for standard atmosphere: - atomic diameter ~ m - distance between molecules ~ 10-8 m - mean free path,  (sea level) ~ 10-7 m   const. 100,000m ;  = m 250,000m ;  = m Knudsen number: Kn = / L  - mean free path L - characteristic length

For continuum assumption: Kn << 1 • Kn < Non-slip fluid flow - B.C.s: no velocity slip - No temp. jump - Classical fluid mechanics • 0.001< Kn < Slip fluid flow - Continuum with slip B.C.s • 0.1< Kn< Transition flow - No continuum, kinetic gas • 10<Kn Free molecular flow Molecular dynamics

1.2.2. Thermodynamical Properties
Thermodynamics - static situation of equilibrium n - mean free time a – speed of molecular motion (~ speed of sound: c) n = /a –microscopic time scale to equilibrium Liquid Gas

Convection time scale s = L / U - L : characteristic length - U : fluid velocity (macroscopic scale) Local thermodynamic equilibrium assumption: n«s - /a « L/U  (/L).(U/a) « 1  Kn.M « 1

Mach number: M = U / a - Incompressible flow: M0, U«a - Compressible flow: - Gas dynamics - M<1 : subsonic - M~1 : transonic - M>1 : supersonic (1<M<5) - M»1 : hypersonic (5<M<40)

1.2.3. Physical Properties Example: density  at point P
•  = density, mass/volume [kg/m3] •  = specific weight [N/m3] =  g • average density in a small volume V = m / V

1.2.3. Physical Properties   = 1000 kg/m3 P ≠ lim(m/V) as V 0
P = lim(m/V) as V  V* V*~=R.E.V. (representative elementary volume) Fluid particle with volume: V*~=(1 m)3 ~109 particles Specific gravity, S.G.: the ratio of a liquid's density to that of pure water at 4oC (39.2oF) 4oC   = 1000 kg/m3 = 1 g/cm3

Physical Properties Similarly, other macroscopic physical properties or physical quantities can be defined from this microscopic viewpoint Momentum M, Velocity u Acceleration a Temperature T Pressure, viscosity, etc…

1.2.4. Force & Acceleration (Newton’s Law)
The force on a body is proportional to the resulting acceleration  F = ma ; unit: 1N = 1kg . 1m/s2 The force of attraction between two bodies is proportional to the masses of the bodies  r = Distance G = Gravitational Constant

Various kinds of forces Static pressure Dynamic pressure Shear force Body force (weight) Surface tension Coriolis force Lorentz force, etc…

Newton’s law is a conservation law. It describes the conservation of linear momentum in a system. Different kinds of conservation Laws, e.g. Conservation of mass Conservation of linear momentum Conservation of energy, etc… Continuity equation Navier-Stokes equations Energy equation, etc…

Viscosity The shear stress on an interface tangent to the direction of flow is proportional to the strain rate (velocity gradient normal to the interface)  = µu/y µ is the (dynamic) viscosity [kg/(m.s)] Kinematic viscosity:  = µ/ [m2/s]

1.2.5. Viscosity Power law:  = k ( u/ y)m
Newtonian fluid: k = µ, m=1 Non-Newtonian fluid: m1 Bingham plastic fluid:  = 0 +µu/y

1.2.5. Viscosity No-slip condition
From observation of real fluid, it is found that it always ‘stick’ to the solid boundaries containing them, i.e. the fluid there will not slip pass the solid surface. This effect is the result of fluid viscosity in real fluid, however small its viscosity may be. A useful boundary condition for fluid problem.

1.2.6. Equation of State (Perfect Gas)
Equation of state is a constitutive equation describing the state of matter Ideal gas: the molecules of the fluid have perfectly elastic collisions Ideal gas law: p = R T R is universal gas constant Speed of sound: c=(dp/d)1/2

Surface Tension At the interface of a liquid and a gas the molecular attraction between like molecules (cohesion) exceed the molecular attraction between unlike molecules (adhesion). This results in a tensile force distributed along the surface, which is the surface tension.

1.2.7. Surface Tension For a liquid droplet in gas in equilibrium:
-(∆p)R2 +  (2R) = 0 ∆p is the inside pressure in the droplet above that of the atmosphere ∆p=pipe = 2 / R

Surface Tension For liquids in contact with gas and solid, if the adhesion of the liquid to the solid exceeds the cohesion in the liquids, then the liquid will rise curving upward toward the solid. If the adhesion to the solid is less than the cohesion in the liquid, then the liquid will be depressed curving downward. These effects are called capillary effects.

Surface Tension The capillary distance, h, depends for a given liquid and solid on the curvature measured by the contact angle , which in turn depends on the internal diameter.  (2R) cos - g(R2)h = 0 → h=2 cos/gR The pressure jump across an interface in general is p =  (1/R1 + 1/R2) For a free surface described by z=x3=η(x1,x2), 1/Ri= ( 2η/ xi2)/[1+( η/ xi)2]3/2

Vapour Pressure When the pressure of a liquid falls below the vapor pressure it evaporates, i.e., changes to a gas. If the pressure drop is due to temperature effects alone, the process is called boiling. If the pressure drop is due to fluid velocity, the process is called cavitation. Cavitation is common in regions of high velocity, i.e., low p such as on turbine blades and marine propellers.

Vapour Pressure

Chapter 2: FLUID STATICS

2.1. Hydrostatic Pressure Fluid mechanics is the study of fluids in macroscopic motion. For a special static case: No Motion at All Recall that by definition, a fluid moves and deforms when subjected to shear stress and, conversely, a fluid that is static (at rest) is not subjected to any shear stress. Otherwise it will move.  No shear stress, i.e., Normal stress only

2.1. Hydrostatic Pressure The force/stress on any given surface immersed in a fluid at rest, is always perpendicular (normal) to the surface. This normal stress is called “pressure” Fluid statics is to determine the pressure field At any given point in a fluid at rest, the normal stress is the same in all directions (hydrostatic pressure)

2.1. Hydrostatic Pressure Proof: Take a small, arbitrary, wedged shaped element of fluid Fluid is in equilibrium, so ∑F = 0 Let the fluid element be sufficiently small so that we can assume that the pressure is constant on any surface (uniformly distributed).

2.1. Hydrostatic Pressure  m =  V  V=  x  y  z/2 F1=p1 A1
Fluid Density :  Fluid Volume :  V=  x  y  z/2

2.1. Hydrostatic Pressure Look at the side view ∑Fx = 0 :
F1 cos - F2 = 0 p1 A1cos - p2 A2 = 0 Since  A1cos =  A2 =  y  z  p1 = p2 ∑Fz = 0 : F1 sin +  m.g = F3 p1  A1 sin +   V g = p3  A3 p1( x/sin )  y sin +  g  x  y  z/2 = p3  x  y p1 +  g  z/2 = p3

2.1. Hydrostatic Pressure Shrink the element down to an infinitesimal point, so that  z0, then p1 = p3 p1 = p2 = p3 Notes: • Normal stress at any point in a fluid in equilibrium is the same in all directions. • This stress is called hydrostatic pressure. • Pressure has units of force per unit area. P = F/A [N/m2] • The objective of hydrostatics is to find the pressure field (distribution) in a given body of fluid at rest.

2.2. Vertical Pressure Variation
Take a fluid element of small control volume in a tank at rest Force balance: (P+  P) A +  g A  y = P A  P/  y = -  g Negative sign indicates that P decreases as y increases A

2.2. Vertical Pressure Variation
For a constant density fluid, we can integrate for any 2 vertical points in the fluid (1) & (2). P2 - P1 = -  g (y2 - y1) If  = (y), then: ∫dP = -g ∫ (y)dy If  = (p,y) such as for ideal gas P =  RT where T=T (y) ∫dP/P = -(g/R) ∫dy/T(y) The integration at the right hand side depends on the distribution of T(y).

2.3. Horizontal Pressure Variation
Take a fluid element of small control volume Force balance: P1A = P2A P1 = P2 Static pressure is constant in any horizontal plane. Having the vertical & horizontal variations, it is possible to determine the pressure at any point in a fluid at rest.

2.3. Horizontal Pressure Variation
Absolute Pressure v.s. Gage Pressure Absolute pressure: Measured from absolute zero Gage pressure: Measured from atmospheric pressure If negative, it is called vacuum pressure Pabs = Patm + Pgage

2.4. Forces on Immersed Surfaces
For constant density fluid: * The pressure varies with depth, P= gh. * The pressure acts perpendicularly to an immersed surface Plane Surface • Let the surface be infinitely thin, i.e. NO volume • Plate has arbitrary plan form, and is set at an arbitrary angle, , with the horizontal.

Looking at the top plate surface only, the pressure acting on the plate at any given h is: P = Patm + gh So, the pressure distribution on the surface is,

To find the total force on the top surface, integrate P over the area of the plate, F = ∫P dA = PatmA + g ∫h dA Note that h = y sin, therefore: F = PatmA + g sin ∫y dA Recall that the location of c.g.(center of gravity) in y is: yc.g. = (1/A) ∫y dA

So, F = PatmA + g sin yc.g.A Or, F = PatmA + ghc.g.A = (Patm + ghc.g)A If Pc.g=Patm + ghc.g , then the pressure acting at c.g. is: F = P c.g. A In a fluid of uniform density, the force on a submerge plane surface is equal to the pressure at the c.g. of the plane multiplied by the area of the plane. F is independent of . The shape of the plate is not important

Where does the total/resultant force act? Similar to c.g., the point on the surface where the resultant force is applied is called the Center of Pressure, c.p.

The moment of the resultant force about the x-axis should equal the moment of the original distributed pressure about the x-axis yc.p.F = ∫y dF =  g sin  ∫y2 dA +Patm ∫y dA Recall that the moment of inertia about the x-axis, Iox, is by definition: Iox = ∫y2 dA = y2c.g.A + Ic.g.x

Ic.g.x - moment of inertia about the x-axis at c.g. yc.p.F = g sin Iox +Patm yc.g.A = g sin (y2c.g.A + Ic.g.x) +Patm yc.g.A =(g sin yc.g.A + PatmA) yc.g.+ g sin Ic.g.x yc.p. = yc.g. + (g sin Ic.g.x) / (Pc.g.A)

Similarly, xc.p. = xc.g. + (g sin Ic.g.y) / (Pc.g.A) Ic.g.y - moment of inertia about the y-axis at c.g. * Tables of Ic.g. for common shapes are available * For simple pressure distribution profiles, the c.p. is usually at "c.g." of the profile

Curved Surface Suppose a warped plate is submerged in water, what is the resulting force on it? The problem can be simplified by examining the horizontal and vertical components separately.

Horizontal Force Zoom on an arbitrary point 'a'. Locally, it is like a flat plate Pa is the pressure acting at 'a', and it is normal to the surface. The force due to the pressure at 'a' is: Fa = Pa Aa, which acts along the same direction as Pa Its horizontal component is: FaH = Fa sin = Pa.Aa sin

But, Aasin is the vertical projection of 'a', so that the horizontal force at 'a' due to pressure is equal to the force that would be exerted on a plane, vertical projection of 'a'. This can be generalized for the entire plane The horizontal force on a curved surface equals the force on the plane area formed by the projection of the curved surface onto a vertical plane The line of action on a curved surface is the same as the line of action on a projected plane

This is true because for every point on the vertical projection there is a corresponding point on the warped plate that has the same pressure.

Vertical Force Similar to the previous approach, FaV = Fa cos = Pa Aacos  Aacos is the horizontal projection of 'a', but this is only at a point! Notice that if one looks at the entire plate, the pressures on the horizontal projection are not equal to the pressures on the plate

Note: Pa= gha  FaV = ghaAa cos In general, Pa ≠ Pa' Consequently, one needs to integrate along the curved plate This is not difficult if the shape of the plate is given in a functional form

The ultimate result is: The vertical component of the force on a curved surface is equal to the total weight of the volume of fluid above it The line of action is through the c.g. of the volume If the lower side of a surface is exposed while the upper side is not, the resulting vertical force is equal to the weight of the fluid that would be above the surface

So far, only surfaces (not volumes) have been discussed In fact, only one side of the surface has been considered Note that for a surface to be in equilibrium, there has to be an equal and opposite force on the other side

2.5. Bodies with Volume (Buoyancy)
The volume can be constructed from two curved surfaces put together, and thus utilize the previous results. Since the vertical projections of both plates are the same, FHab = FHcd, Where FVab =g (vol. 1-a-b-2-1), FVcd =g (vol. 1'-d-c-2'-1') *Note that this is true regardless of whether there is or there isn't any fluid above c-d.

2.5. Bodies with Volume (Buoyancy)
Join the two plates together Total force: FB=FVcd-FVab= g(vol. a-b-c-d) This force FB is called Buoyancy Force

2.6. Archimedes' Principle The net vertical force on an immersed body of arbitrary shape due to the pressure forces acting on the surfaces of the body is equal to the weight of the displaced fluid * The line of action is through the center of the mass of the displaced fluid volume * Direction of buoyant force is upward If a body immersed in a fluid is in equilibrium, then: W = FB W is the weight of the body.

2.6. Archimedes' Principle For a body in a fluid of varying density, e.g. ocean, the body will sink or rise until it is at a height where its density is equal to the density of the fluid For a body in a constant density fluid, the body will float at a level such that the weight of the volume of fluid it displaces is equal to its own weight

2.7. Pressure Variation with Rigid-Body Motion
The variation of pressure with distance is balanced by the total accelerations that may be due to gravitational acceleration g, constant linear acceleration al and constant rotational acceleration ar. Generally, a = -(g + al + ar) For g in the vertical y direction, g = gj For linear acceleration in the x and y directions, al = axi + ayj For fluid rotates rigidly at a constant angular velocity ω, the acceleration ar is in the radial r direction, i.e., ar = -rω2er where er is the unit vector in r direction

Chapter 3: FLUID IN MOTIONS

3.1. Newton’s Second Law The net force F acting on a matter of mass m leads to an acceleration a following the linear relation: F = ma For a solid body of fixed shape, m is a constant and a is described along the trajectory of motion.

3.1. Newton’s Second Law If r(t) represents the particle trajectory, the velocity v(t) and acceleration are then given by: v(t) = dr/dt ; a(t) = dv/dt = d2r/dt2 Therefore, F = mdv/dt If m = m(t), then F = (mdv/dt)+(vdm/dt)=d(mv)/dt = dM/dt. where M = mv is the momentum.

3.1. Newton’s Second Law For fluids enclosed in a control volume V(t) which may deform with time along the trajectory of motion, it is only correct to use the fluid momentum M to describe the Newton’s second law For M= , we have F = dM/dt =

3.2. Description of Fluid Flow
Fluid dynamics is the mechanics to study the evolution of fluid particles in a space domain (flow field). There are two ways to describe the flow field Lagrangian description Eulerian description

3.2.1 Lagrangian Description
Given initially the locations of all the fluid particles, a Lagrangian description is to follow historically each particle motion by finding the particle locations and properties at every time instant.

3.2.1 Lagrangian Description
Therefore, if a specific fluid particle is initially (t=t0) located at (x0, y0, z0), the Lagragian description is to determine (x(t), y(t), z(t)) and the fluid properties, such as f(t) = f [x(t), y(t), z(t); t], v(t) = v [x(t), y(t), z(t); t] , etc. given f0 = f (x0, y0, z0; t0), v0 = v (x0, y0, z0; t0) etc. Note that (x(t), y(t), z(t)) is a function of time. (for any function f)

Eulerian Description An Eulerian description of fluid flow is simply to state the evolution of fluid properties at a fixed point (x, y, z) with time. Here (x, y, z) is independent of time. Hence, f(t) = f[x, y, z, t] , v = v [x, y ,z, t] etc. Eulerian description is to observe the fluid properties of different fluid particles passing through the same fixed location at different time instant, while Lagrangian description is to observe the fluid properties at different locations following the same particle

3.2.3. Relation between Lagrangian and Eulerian description
It is important to note that there is only one flow property at the same location with respect to the same time, i.e., f(x(t),y(t),z(t),t) = f(x,y,z,t). Therefore, the Lagrange differential with respective to dt, which is equal to the total Eulerian differential:

3.2.3. Relation between Lagrangian and Eulerian description
Hence, the total derivative is given by:

3.3. Equations of Motion for Inviscid Flow
Conservation of Mass Conservation of Momentum

Conservation of Mass Mass in fluid flows must conserve. The total mass in V(t) is given by: Therefore, the conservation of mass requires that dm/dt = 0. where the Leibniz rule was invoked.

3.3.1. Conservation of Mass Hence:
This is the Integral Form of mass conservation equation.

Conservation of Mass Integral form of mass conservation equation By Divergence theorem: Hence: = 0

Conservation of Mass As V(t)→0, the integrand is independent of V(t) and therefore, This is the Differential Form of mass conservation and also called as continuity equation.

3.3.2. Conservation of Momentum
The Newton’s second law, is Lagrangian in a description of momentum conservation. For motion of fluid particles that have no rotation, the flow is termed irrotational. An irrotational flow does not subject to shear force, i.e., pressure force only. Because the shear force is only caused by fluid viscosity, the irrotational flow is also called as “inviscid” flow

For fluid subjecting to earth gravitational acceleration, the net force on fluids in the control volume V enclosed by a control surface S is: where s is out-normal to S from V and the divergence theorem is applied for the second equality. This force applied on the fluid body will leads to the acceleration which is described as the rate of change in momentum.

where the Leibniz rule was invoked.

Hence: This is the Integral Form of momentum conservation equation.

Integral form of momentum conservation equation By Divergence theorem: Hence:

As V→0, the integrands are independent of V. Therefore, This is the Differential Form of momentum conservation equation for inviscid flows.

By invoking the continuity equation, The momentum equation can take the following alternative form: which is commonly referred to as Euler’s equation of motion.

3.4. Bernoulli Equation for Steady Flows
Bernoulli equation is a special form of the Euler’s equation along a streamline. For a first look, we restrict our discussion to steady flow so that the Euler’s equation becomes: Assuming that g is in the negative z direction, i.e., g = and using the following vector identity, the Euler’s equation for steady flows becomes

We now take the scalar product to the above equation by the position increment vector dr along a streamline and observe that Thus, the result leads to The above equation now can be integrated to give ; (for any function f) (along streamline)

For incompressible fluids where ρ = constant, we have For irrotational flows, everywhere in the flow domain and (along streamline)

Since, for dr in any direction, we have: For anywhere of irrotational fluids For anywhere of incompressible fluids

3.5. Static, Dynamic, Stagnation and Total Pressure
Consider the Bernoulli equation, The static pressure ps is defined as the pressure associated with the gravitational force when the fluid is not in motion. If the atmospheric pressure is used as the reference for a gage pressure at z=0. (for incompressible fluid)

Then we have as also from chapter 2. The dynamic pressure pd is then the pressure deviates from the static pressure, i.e., p = pd+ps. The substitution of p = pd+ps. into the Bernoulli equation gives

The maximum dynamic pressure occurs at the stagnation point where v=0 and this maximum pressure is called as the stagnation pressure p0. Hence, The total pressure pT is then the sum of the stagnation pressure and the static pressure, i.e., pT= p0 - ρgz. For z = -h, the static pressure is ρgh and the total pressure is p0 + ρgh.

3.6. Energy Line and Hydraulic Grade Line
In fact the Bernoulli equation also states that the energy density (per unit volume) possessed by the fluid particle is constant not only along a streamline but also at everywhere in fluid domain for irrotational flow. The energy consists of pressure energy (p), kinetic energy (ρv2/2) and gravitational potential energy (ρgz).

It becomes simpler if this total energy is interpreted into a total head H (height from a datum) by dividing the Bernoulli equation with ρg such that where p/ρg is the pressure head, v2/2g is the velocity head and z is the elevation head.

A piezometric head is then defined as that consists of only the pressure and elevation heads, i.e., The variations of H and Hp along the path of fluid flow can be plotted into lines and are termed as “energy line” and “hydraulic grade line”, respectively. It is noted that H is always higher than Hp and that a negative pressure (below atmospheric pressure) occurs when Hp is below the fluid streamline

3.7. Applications of Bernoulli Equation
Pitot-Static Tube Free Jets Flow Rate Meter

Pitot-Static Tube Pitot-static tube is a device that measures the difference between h1 and h2 so that the velocity of the fluid flow at the measurement location can be determined from

Pitot-Static Tube

Free Jets Free jets are the flow from an orifice of an apparatus that converts the total elevation head h into velocity head, i.e.,

Flow Rate Meter The commonly used flow rate meter is the Venturi meter that determines the flow rate Q through pipes by measuring the difference of piezometric heads at locations of different cross-sectional areas along the pipe.

Flow Rate Meter If hp1 and hp2 represent the piezometric heads at section 1 and 2 with cross-sectional areas A1 and A2 respectively, we have: and Then, Therefore,

Chapter 4: FLUID KINETMATICS

4. FLUID KINETMATICS Fluid kinematics concerns the motion of fluid element. As the fluid flows, a fluid particle (element) can translate, rotate, and deform linearly and angularly Translation Rotation Linear deformation Circulation Dilation Viscous stress Angular deformation

4.1. Translation The translation considers mainly the velocity and acceleration along the trajectory of fluid element in linear motion z y x

4.1. Translation which, for steady flows, reduces to
For the fluid element moving along the trajectory r(t), the velocity is simply given by v =dr/dt = (u,v,w). As the description is basically Lagrangian, the acceleration a is given by which, for steady flows, reduces to

4.2. Linear Deformation (Strain)
Deformation: change of shape of fluid element For easily understanding, we illustrate here in two-dimensions. The results then can be easily extended to 3-dimensions. Consider the rectangular fluid element at the initial time instant given in the following picture

The initial distance between points A and B is ∆x and between A and C is ∆y. After a short time of ∆t, the distances then become ∆x+∆Lx and ∆y+∆Ly due to different velocities at B and C from A

The linear strain rate in x and y directions are then given by Similarly, for 3-D flows we have in the z-direction,

4.3. Dilation Volumetric expansion & contraction
The fluid dilation is defined as the change of volume per unit volume. We are more interested in the rate of dilation that determines the compressibility of fluids. For 2-D flows,

4.3. Dilation Then, the rate of dilatation becomes, for 2-D flows
It is easy to generalize this dilation rate for 3-D flows and to reach For incompressible flow, the rate of dilation is zero, for 2-D flows

4.4. Angular Deformation (Strain)
Now consider the deformation between A and B caused by the change in velocity v, and the deformation between A and C by change in u

For , the counter clockwise rotation of AB is equal to clockwise rotation of AC; therefore, the fluid element is in pure angular strain without net rotation and the angular strain is equal to either or However, if ≠ , the strain then is equal to . The rate of angular strain is then given by

Similarly, we can extend to other planes y-z and z-x to obtain:

4.5. Rotation If then the fluid element is under rigid body rotation on the x-y plane. No angular strain is experienced, i.e., with

4.5. Rotation When ≠ , the rotation of fluid element in x-y plane is the average rotation of the two mutually perpendicular lines AB and AC; therefore, where a counter clockwise rotation is chosen as positive and the rotation axis is in the z direction

4.5. Rotation          y u x v Ω w z 2 1 ,
Rotation is a vector quantity for fluid elements in 3-D motion. A fluid particle moving in a general 3-D flow field may rotate about all three coordinate axes, thus: and so,          y u x v Ω w z 2 1 ,

4.5. Rotation The vorticity of a flow field is defined as

4.5. Rotation Therefore, The flow vorticity is twice the rotation
In 2-D flow, ∂/∂z=0 and w=0 (or const.), so there is only one component of vorticity, Irrotational flow is defined as having

4.5. Rotation A fluid particle moving, without rotation, in a flow field cannot develop a rotation under the action of a body force or normal surface force. If fluid is initially without rotation, the development of rotation requires the action of shear stresses. The presence of viscous forces implies the flow is rotational The condition of irrotationality can be a valid assumption only when the viscous forces are negligible. (as example, for flow at very high Reynolds number, Re, but not near a solid boundary)

4.6. Circulation Consider the flow field as shown below
The circulation, , is defined as the line integral of the tangential velocity about a closed curve fixed in the flow, where ds is the tangential vector along the integration loop. i.e with being the unit tangential vector

4.6. Circulation Where is the line-element vector tangent to the closed loop C of the integral. It is possible to decompose the integral loop C into the sum of small sub-loops, i.e., Without loss of generality, each sub-loop can be a rectangular grid as illustrated below.

4.6. Circulation Therefore, As a result, we have
where A is the area enclosed the contour

4.6. Circulation Stokes' theorem in 2-D:
The circulation around a closed contour (loop) is the sum of the vorticity (flux) passing through the loop This is an expression to illustrate the Green’s Theorem. In fact, the surface A can be a curved surface

4.6. Circulation Then for each sub-loop on the surface, we have locally where is the vorticity normal to the surface enclosed by the small increment loop C

4.7. Viscous Stresses The strain rate tensor S is a symmetric tensor that measures the rate of linear and angular deformations of fluid element. The strain rate tensor is expressed as: where the superscript “T” represents the transpose

4.7. Viscous Stresses In term of a Cartesian coordinate system, they are expressed as: and

4.7. Viscous Stresses Following the Stokes’ hypothesis, the viscous stress tensor is linearly related to the rate of dilation and the strain rate tensor by where I represents the unit tensor, i.e.,

4.7. Viscous Stresses The proportional constants of the above linear relation are the volume viscosity and shear viscosity of the fluid respectively. It is seen that the fluid viscosity leads to additional normal stresses, as well as shear stresses. Note that is a symmetric tensor, i.e., Total stress is given by

Chapter 5: EQUATIONS OF MOTION OF VISCOUS FLOWS

5.1 Flow Fields and Gradients
The continuum assumption allows the treatment of fluid properties as fields, scalar vector or tensor, which are function of space (r) and time (t) Scalar fields: density – pressure – temperature – Vector fields: velocity – vorticity – Tensor fields: total stress – viscous stress – strain rate –

Total change of a scalar field, , due to change in space only: For the position vector given by , the change in is only cause by the change in r described by

The total derivative is then given by where is the gradient of . The gradient is a vector along the direction where the magnitude of has a maximum.

Consider P & Q to be 2 points on a surface where These points are chosen so that Q is a small distance from P. Then dr is tangential to the surface. Now let’s move from P to Q. The change in is then given by, since the 2 points are on the surface with the same C.

Therefore, is perpendicular to dr from P. Since dr may be in any direction from P, as long as it is tangential to the surface , we conclude that has to be in normal to the surface

Example: For unsteady flows where may change with time, recall that the total derivative is

5.2 Conservation of Mass We recall in Chapter 3 that the conservation of mass can be understood more easily in the Lagrangian frame. It states that the total mass m in control volume V has to be conserved if the control volume deformed with the flow to confine the same fluid particles.

5.2 Conservation of Mass We now extend further to include the cases when there is a mass source in the control volume. If represents the rate of mass source per unit volume, the mass balance then read:

5.2 Conservation of Mass Above equation is the integral form of the mass conservation in the Eulerian description. Note that the control volume V can be either fixed or varying. The first term on the left hand side is the contribution caused by the density change in V and the second term caused by the mass flux enter the surface that define the control volume. The term on the right hand side then represents the rate of mass created or annihilated in V

5.2 Conservation of Mass To obtain the differential form, we now employ the divergence theorem to the second term on the left of conservation of mass equation to give:

5.2 Conservation of Mass Again, as V(t)→0 the integrand is independent of V and therefore, which is the differential form of equation for mass conservation. Note that in the above equation, all terms are in rate of mass change per unit volume.

5.2 Conservation of Mass For flow fields without mass sources, the integral and differential forms of conservation of mass equation reduce to and respectively, which were given in Chapter 3

5.2.1 Derivation of the Differential Equation in Eulerian frame and Cartesian Coordinate
Consider the fixed control volume as shown below:

Net mass leaving the control volume/time

Net mass increase in the control volume/time
5.2.1 Derivation of the Differential Equation in Eulerian frame and Cartesian Coordinate Net mass increase in the control volume/time Conservation of mass states that the net mass entering the control volume/unit time is equal to the rate of increase of mass in the differential control volume

5.2.2 Special Cases Steady flow, Incompressible flow, 2-D flow,
Cartesian: Polar: Spherical: 2-D flow,

5.3 Conservation of Momentum (Navier-Stokes Equations)
In Chapter 3, for inviscid flows, only pressure forces act on the control volume V since the viscous forces (stress) were neglected and the resultant equations are the Euler’s equations. The equations for conservation of momentum for inviscid flows were derived based on Newton’s second law in the Lagrangian form.

Here we should include the viscous stresses to derive the momentum conservation equations. With the viscous stress, the total stress on the fluid is the sum of pressure stress( , here the negative sign implies that tension is positive) and viscous stress ( ), and is described by the stress tensor given by:

Here, we generalize the body force (b) due to all types of far field forces. They may include those due to gravity , electromagnetic force, etc. As a result, the total force on the control volume in a Lagrangian frame is given by

The Newton’s second law then is stated as: Hence, we have

By the substitution of the total stress into the above equation, we have which is integral form of the momentum equation.

For the differential form, we now apply the divergence theorem to the surface integrals to reach: Hence, V→0, the integrands are independent of V. Therefore, which are the momentum equations in differential form for viscous flows. These equations are also named as the Navier-Stokes equations.

For the incompressible fluids where = constant. If the variation in viscosity is negligible (Newtonian fluids), the continuity equation becomes , then the shear stress tensor reduces to

The substitution of the viscous stress into the momentum equations leads to: where is the Laplacian operator which in a Cartesian coordinate system reads

For inviscid flow where , the above equation reduces to the Euler’s equation given in Chapter 3 where the body force is also taken the form due to gravity.

Chapter 6: DIMENTIONAL ANALYSIS

6 Dimensional Analysis The objective of dimensional analysis is to obtain the key non- dimensional parameters that govern the physical phenomena of flows. Variables like etc. are combined to form the key parameters that have no physical units (dimensionless). The non-dimensional parameters include both the geometric and dynamic parameters.

6.1 Similarity Geometric Similarity Dynamic Similarity
Two flows are said to be similar if they have the same geometric and dynamic dimensionless parameters.

6.1.1 Geometrical Similarity
Two body are geometrically similar, if the geometry of one can be obtained from another by scaling all dimensions by the same factor. B1 A1 A A2 B B2 C2 C C1

6.1.2 Dynamic Similarity Flows are said to be dynamically similar if by scaling the dependent and independent variables they yield the same non-dimensional parameter. This is more difficult to achieve than the geometric similarity. What are these non-dimensional parameters? How can they be found?

6.2 Buckingham Pi Theorem Dimensionless product is the product of several dimensional quantities that render the product dimensionless The rank of matrix Nm is the minimum dimensions that leads to non-zero determinant, which is also the minimum dimensions of the quantities that describe the physics.

6.2 Buckingham Pi Theorem Given the quantities that are required to describe a physical law, the number of dimensionless product (the “Pi’s”, Np) that can be formed to describe the physics equals the number of quantities (Nv) minus the rank of the quantities, i.e., Np=Nv – Nm,

6.2.1 Example Viscous drag on an infinitely long circular cylinder in a steady uniform flow at free stream of an incompressible fluid. Geometrical similarity is automatically satisfied since the diameter (R) is the only length scale involved. D D: drag (force/unit length)

6.2.1 Example Dynamics similarity

6.2.1 Example Both sides must have the same dimensions!

6.2.1 Example The non-dimensional parameters are:
where is the kinematic viscosity. The non-dimensional parameters are:

6.2.1 Example Therefore, the functional relationship must be of the form: The number of dimensionless groups is Np=2

6.2.1 Example The matrix of the exponents is
The rank of the matrix (Nm) is the order of the largest non-zero determinant formed from the rows and columns of a matrix, i.e. Nm=3.

6.2.1 Example Problems: No clear physics can be based on to know the involved quantities Assumption is not easy to justified.

6.3 Normalization Method The more physical method for obtaining the relevant parameters that govern the problem is to perform the non-dimensional normalization on the Navier-Stokes equations: where the body force is taken as that due to gravity.

6.3 Normalization Method As a demonstration of the method, we consider the simple steady flow of incompressible fluids, similar to that shown above for steady flows past a long cylinder. Then the Navier-Stokes equations reduce to:

6.3 Normalization Method If the proper scales of the problem are:
U L P: If the proper scales of the problem are: Here the flow domain under consideration is assumed such that the scales in x, y and z directions are the same

6.3 Normalization Method Using these scales, the variables are normalized to obtain the non-dimensional variables as: Note that the non-dimensional variable with “*” are of order one, O(1). The velocity scale U and the length scale L are well defined, but the scale P remains to be determined.

6.3 Normalization Method The Navier-Stokes equations then become:
where is the unit vector in the direction of gravity which is dimensionless.

6.3 Normalization Method The coefficient of in the continuity equation can be divided to yield Dividing the momentum equations by in the first term of left hand side gives:

6.3 Normalization Method Since the quantities with “*” are of O(1), the coefficients appeared in each term on the right hand side measure the ratios of each forces to the inertia force. i.e., where Re is called as Reynolds number and Fr as Froude number.

6.3 Normalization Method The dynamic of fluid motion then depends solely on the magnitudes of these non-dimension parameters, i.e., pressure coefficient and gravitational body-force coefficient. Flows are dynamically similar if they have the same Re and Fr

6.3 Normalization Method Remark:
New non-dimensional parameters can also emerge from the non-dimensional analysis on the boundary conditions which is not deliberated here. The dimensional analysis reduces experimentalists the need of carrying out measurements for different U, D, etc.

6.4 Characteristics of Non-Dimensional Parameters
Reynolds Number Froude Number

6.4.1 Reynolds Number Let’s for simplicity consider the case where the gravitational force has no consequence to the dynamic of the flow, i.e. the case where or the contribution of is only to the static pressure. Then, the pressure P represents the dynamics pressure. The normalized momentum equation becomes

6.4.1 Reynolds Number where is the viscous diffusion length in an advection time interval of Here, measures the time required for fluid travel a distance L.

6.4.1 Reynolds Number High Reynolds Number Flow, Re>>1
Intermediate Reynolds Number Flow, Re~1 Low Reynolds Number Flow, Re<<1

6.4.1.1 High Reynolds Number Flow
When , inertia force is much greater than viscous force, i.e., the viscous diffusion distance is much less than the length L. Viscous force is unimportant in the flow region of , but can become very important in the region of near the solid boundary. This flow region near the solid boundary is called an boundary layer as first illustrated by Prandtl.

Flow in the region outside the boundary layer where viscous force is negligible is inviscid. The inviscid flow is also called the potential flow. U Boundary layer flow Potential flow

The normalized dimensionless equation to the first order approximation is: Clearly, the proper pressure scale should be chosen such that is of O(1), and for simplicity, can be set as is the proper pressure scale for high Reynolds number flow. Flows in the boundary layer are governed by boundary layer equations that need to be derived separately with different approach

For inviscid, incompressible, steady flow, the governing equations in terms of dimensional variable to the first order approximation are written as, If the gravitational force is retrieved, then we have where is the steady Euler equations for incompressible fluid as given in Chapter 2.

6.4.1.2 Intermediate Reynolds Number Flow
When , inertia forces and viscous forces are of equal importance. The flow is viscous in a region of surrounding the body since . No approximation can be done. L U

6.4.1.2 Intermediate Reynolds Number Flow
The governing equations remain as: whose solutions satisfying proper boundary conditions can usually only be obtained numerically.

6.4.1.3 Low Reynolds Number Flow
When , the inertia force is very much smaller than the viscous force. The viscous diffusion length is much larger than L. The flow is viscous for almost the entire region except at vary far away from the solid boundary.

Since implies , the inertia force is negligible and the pressure force has to balance the viscous force. Therefore, the proper scale for P is such that The governing equation for low Reynolds number flows, without the gravitational force, can be approximated to the first order by

If the gravitational force is recapped, the governing equations to the first order approximation becomes: Flows of low Reynolds number are called the Stokes flows, or creeping flows. One such example is the settling of small particles in water. Lava flows from volcanic eruption are also typical low Reynolds number flows although the viscosity is non-Newtonian.

6.4.1.4 Dynamically Similar & Reynolds Number
Flows with the same Reynolds number are dynamically similar. For example, flight of very small insects in air can be studied much more easily on large models in very viscous fluid (liquids). Similarly, flows for large object such as train, airplane, tall buildings, etc., can be studied experimentally with small models. (Note: They should be geometrically similar)

6.4.2 Froude Number The Froude number measures the gravitational effect on the flows. It depends on the problem encountered. For instance in free surface flows, is the phase speed c of shallow water gravity waves when L is water depth. The Froude number becomes:

6.4.2 Froude Number If U represents the speed of a ship moving on a flat water surface, the wave patterns generated by the ship, called “ship wakes”, then depend on whether the Froude number is U c Fr >1 Fr =0 Fr <1 Ship wakes

6.4.2 Froude Number Another example is the rising of a hot-air balloon, which is characterized by the net buoyancy force. The natural convection flows generated by a vertical heated surface represents another example.  g

6.4.2 Froude Number The ratio of the net buoyancy force (per unit volume) to the inertia force (per unit volume) is: where Ri is the Richardson number. The Richardson number is the key parameter in dealing with advection of fluid of different density. Another example is the propagation internal waves in stratified fluids.

6.5 Map based on Non-Dimensional Parameter Flow
ln Fr Re~1 Fr~1 ln Re Fr~0 Re~0

Chapter 7: INVISCID FLOWS

7.1 Inviscid Flow Inviscid flow implies that the viscous effect is negligible. This occurs in the flow domain away from a solid boundary outside the boundary layer at Re. The flows are governed by Euler Equations where , v, and p can be functions of r and t .

7.1 Inviscid Flow On the other hand, if flows are steady but compressible, the governing equation becomes where  can be a function of r For compressible flows, the state equation is needed; then, we will require the equation for temperature T also.

7.1 Inviscid Flow Compressible inviscid flows usually belong to the scope of aerodynamics of high speed flight of aircraft. Here we consider only incompressible inviscid flows. For incompressible flow, the governing equations reduce to where  = constant.

7.1 Inviscid Flow For steady incompressible flow, the governing eqt reduce further to where  = constant. The equation of motion can be rewrited into Take the scalar products with dr and integrate from a reference at  along an arbitrary streamline =C , leads to since

7.1 Inviscid Flow If the constant (total energy per unit mass) is the same for all streamlines, the path of the integral can be arbitrary, and in the flow domain except inside boundary layers. Finally, the governing equations for inviscid, irrotational steady flow are Since is the vorticity , flows with are called irrotational flows.

7.1 Inviscid Flow Note that the velocity and pressure fields are decoupled. Hence, we can solve the velocity field from the continuity and vorticity equations. Then the pressure field is determined by Bernoulli equation. A velocity potential  exists for irrotational flow, such that, and irrotationality is automatically satisfied.

7.1 Inviscid Flow The continuity equation becomes which is also known as the Laplace equation. Every potential satisfy this equation. Flows with the existence of potential functions satisfying the Laplace equation are called potential flow.

7.1 Inviscid Flow The linearity of the governing equation for the flow fields implies that different potential flows can be superposed. If 1 and 2 are two potential flows, the sum =(1+2) also constitutes a potential flow. We have However, the pressure cannot be superposed due to the nonlinearity in the Bernoulli equation, i.e.

7.2 2D Potential Flows If restricted to steady two dimensional potential flow, then the governing equations become E.g. potential flow past a circular cylinder with D/L <<1 is a 2D potential flow near the middle of the cylinder, where both w component and /z0. U L y x z D

7.2 2D Potential Flows The 2-D velocity potential function gives and then the continuity equation becomes The pressure distribution can be determined by the Bernoulli equation, where p is the dynamic pressure

7.2 2D Potential Flows For 2D potential flows, a stream function (x,y) can also be defined together with (x,y). In Cartisian coordinates, where continuity equation is automatically satisfied, and irrotationality leads to the Laplace equation, Both Laplace equations are satisfied for a 2D potential flow

7.2 Two-Dimensional Potential Flows
For two-dimensional flows, become: In a Cartesian coordinate system In a Cylindrical coordinate system and and

Therefore, there exists a stream function such that in the Cartesian coordinate system and in the cylindrical coordinate system. The transformation between the two coordinate systems

The potential function and the stream function are conjugate pair of an analytical function in complex variable analysis. The conditions: These are the Cauchy-Riemann conditions. The analytical property implies that the constant potential line and the constant streamline are orthogonal, i.e., and to imply that

7.3 Simple 2-D Potential Flows
Uniform Flow Stagnation Flow Source (Sink) Free Vortex

7.3.1 Uniform Flow For a uniform flow given by , we have Therefore,
Where the arbitrary integration constants are taken to be zero at the origin. and and

7.3.1 Uniform Flow This is a simple uniform flow along a single direction.

7.3.2 Stagnation Flow For a stagnation flow, Hence, Therefore,

7.3.2 Stagnation Flow The flow an incoming far field flow which is perpendicular to the wall, and then turn its direction near the wall The origin is the stagnation point of the flow. The velocity is zero there. x  y

7.3.3 Source (Sink) Consider a line source at the origin along the z-direction. The fluid flows radially outward from (or inward toward) the origin. If m denotes the flowrate per unit length, we have (source if m is positive and sink if negative). Therefore,

7.3.3 Source (Sink) The integration leads to and
Where again the arbitrary integration constants are taken to be zero at and

7.3.3 Source (Sink) A pure radial flow either away from source or into a sink A +ve m indicates a source, and –ve m indicates a sink The magnitude of the flow decrease as 1/r z direction = into the paper. (change graphics)

7.3.4 Free Vortex Consider the flow circulating around the origin with a constant circulation . We have: where fluid moves counter clockwise if is positive and clockwise if negative. Therefore,

7.3.4 Free Vortex The integration leads to where again the arbitrary integration constants are taken to be zero at and

7.3.4 Free Vortex The potential represents a flow swirling around origin with a constant circulation . The magnitude of the flow decrease as 1/r.

7.4. Superposition of 2-D Potential Flows
Because the potential and stream functions satisfy the linear Laplace equation, the superposition of two potential flow is also a potential flow. From this, it is possible to construct potential flows of more complex geometry. Source and Sink Doublet Source in Uniform Stream 2-D Rankine Ovals Flows Around a Circular Cylinder

7.4.1 Source and Sink Consider a source of m at (-a, 0) and a sink of m at (a, 0) For a point P with polar coordinate of (r, ). If the polar coordinate from (-a,0) to P is and from (a, 0) to P is Then the stream function and potential function obtained by superposition are given by:

7.4.1 Source and Sink

7.4.1 Source and Sink Hence, Since We have

7.4.1 Source and Sink We have By Therefore,

7.4.1 Source and Sink The velocity component are:

7.4.1 Source and Sink

7.4.2 Doublet The doublet occurs when a source and a sink of the same strength are collocated the same location, say at the origin. This can be obtained by placing a source at (-a,0) and a sink of equal strength at (a,0) and then letting a  0, and m , with ma keeping constant, say 2am=M

7.4.2 Doublet For source of m at (-a,0) and sink of m at (a,0)
Under these limiting conditions of a0, m , we have

7.4.2 Doublet Therefore, as a0 and m with 2am=M
The corresponding velocity components are

7.4.2 Doublet

7.4.3 Source in Uniform Stream
Assuming the uniform flow U is in x-direction and the source of m at(0,0), the velocity potential and stream function of the superposed potential flow become:

The velocity components are: A stagnation point occurs at Therefore, the streamline passing through the stagnation point when The maximum height of the curve is

For underground flows in an aquifer of constant thickness, the flow through porous media are potential flows. An injection well at the origin than act as a point source and the underground flow can be regarded as a uniform flow.

D Rankine Ovals The 2D Rankine ovals are the results of the superposition of equal strength sink and source at x=a and –a with a uniform flow in x-direction. Hence,

D Rankine Ovals Equivalently,

7.4.4 2-D Rankine Ovals The stagnation points occur at
where with corresponding

7.4.4 2-D Rankine Ovals which can only be solved numerically.
The maximum height of the Rankine oval is located at when ,i.e., which can only be solved numerically.

D Rankine Ovals rs ro

7.4.5 Flows Around a Circular Cylinder
Steady Cylinder Rotating Cylinder Lift Force

Steady Cylinder Flow around a steady circular cylinder is the limiting case of a Rankine oval when a0. This becomes the superposition of a uniform parallel flow with a doublet in x-direction. Under this limit and with M=2a. m=constant, is the radius of the cylinder.

Steady Cylinder The stream function and velocity potential become: The radial and circumferential velocities are:

Steady Cylinder ro

Rotating Cylinder The potential flows for a rotating cylinder is the free vortex flow given in section Therefore, the potential flow of a uniform parallel flow past a rotating cylinder at high Reynolds number is the superposition of a uniform parallel flow, a doublet and free vortex. Hence, the stream function and the velocity potential are given by

Rotating Cylinder The radial and circumferential velocities are given by

Rotating Cylinder The stagnation points occur at From

Rotating Cylinder

7.4.5.2 Rotating Cylinder The stagnation points occur at Case 1:

Rotating Cylinder Case 1:

7.4.5.2 Rotating Cylinder Case 2:
The two stagnation points merge to one at cylinder surface where

7.4.5.2 Rotating Cylinder Case 3:
The stagnation point occurs outside the cylinder when where The condition of leads to Therefore, as , we have

Rotating Cylinder Case 3:

Lift Force The force per unit length of cylinder due to pressure on the cylinder surface can be obtained by integrating the surface pressure around the cylinder. The tangential velocity along the cylinder surface is obtained by letting r=ro,

Lift Force The surface pressure as obtained from Bernoulli equation is where is the pressure at far away from the cylinder.

Lift Force Hence, The force due to pressure in x and y directions are then obtained by

Lift Force The development of the lift on rotating bodies is called the Magnus effect. It is clear that the lift force is due to the development of circulation around the body. An airfoil without rotation can develop a circulation around the airfoil when Kutta condition is satisfied at the rear tip of the air foil. Therefore, The tangential velocity along the cylinder surface is obtained by letting r=ro: This forms the base of aerodynamic theory of airplane.

Chapter 8: BOUNDARY LAYER FLOWS

8.1 Boundary Layer Flow The concept of boundary layer is due to Prandtl. It occurs on the solid boundary for high Reynolds number flows. Most high Reynolds number external flow can be divided into two regions: Thin layer attached to the solid boundaries where viscous force is dominant, i.e. boundary layer flow region. Other encompassing the rest region where viscous force can be neglected, i.e., the potential flow region, that has been discussed in chapter 7.

8.1 Boundary Layer Flow The thin layer adjacent to a solid boundary is called the boundary layer and the flow inside the layer is called the boundary layer flow Inside the thin layer the velocity of the fluid increases from zero at the wall (no slip) to the full value of corresponding potential flow. There exists a leading edge for all external flows. The boundary layer flow developing from leading edge is laminar

8.2 Boundary Layer Equations
For simplicity of illustration, we shall consider an incompressible steady flow over a semi-infinite flat plate with an uniform incoming flow of velocity U in parallel to the plate. The flow is two dimensional. The coordinates are chosen such that x is in the incoming flow direction with x=0 being located the leading edge and y is normal to the plate with y=0 being located at the plate wall.

The continuity and Navier-Stokes equations read:

The above equations apply generally to two dimensional steady incompressible flows for all Reynolds number over the entire flow domain. We now seek the equations that provide the first order approximation for high Reynolds number flows in the boundary layer.

When normalize based on the following scales, we recall the normalized governing equations with Re underneath the viscous term

When the viscous terms are dropped for high Re number flows, the equations become those for potential flows outside the boundary layer. The boundary layer effect is not realized. Using L to normalize y cannot resolve the boundary layer near the solid boundary. We need to choose a proper length scale to normalize the y coordinate.

To this end, let L be the characteristic length in the x direction and that L be sufficiently long, such that Therefore, the viscous diffusion layer thickness L at x=L is small compared to L, i.e., This viscous diffusion layer near the wall is the boundary layer.

To resolve the flow in the boundary layer, the proper length scale in y- direction is L while that in x-direction remains as L. The condition of v=0 for potential flows near the wall outside the boundary layer and the continuity Equation also imply that the velocity v in the boundary layer is small compared to U. Let V be the scale of v in the boundary layer, then V<<U. It is clear that the non-dimensional normalized variables can now be expressed as:

For high Reynolds number flow, the proper pressure scale is ; hence, In terms of the dimensionless variables, the governing equations becomes:

From the continuity equation, we need such that Therefore, , and the substitution of V into the momentum equation leads to:

In order to balance the shear force with the inertia force, it is clear that we need, ,i.e., The momentum equations reduced further to For high Reynolds number flows, the terms with ReL to the first approximation can be neglected.

These results in the boundary layer equations that in dimensional form are given by: Continuity: X-momentum: Y-momentum:

The last equation for y-momentum equation indicates that the pressure is constant across the boundary layer, i.e., equal to that outside the boundary layer (in the free stream), i.e., In the free stream (outside the boundary layer), the viscous force is negligible and we also have , which in fact is the slip velocity of corresponding potential theory near the boundary The x-momentum boundary layer equation near the free stream becomes:

Therefore, the boundary layer equations can be re-written into: and the proper boundary conditions are:

For semi-infinite flat plate with uniform incoming velocity, The boundary layer equations reduced further to:

8.3 Boundary Layer Flows over Curve Surfaces
In fact the boundary layer equation is also meant for curved solid boundary, given a large radius of curvature R >> L.

By defining an orthogonal coordinate system with x coordinate along boundary and y coordinate normal to boundary, previous analysis is also valid for curved surface. This can be done through a coordinate transformation. Since radius of curvature is large, the curvature effects become higher order terms after transformation. These higher order terms can be neglected for 1st-order approximation. The same boundary layer equation can be obtained.

For example in 2D flows, one way is to use the potential lines and streamlines to form a coordination system. x is along streamline direction, and y is the along potential lines. Such coordination system are called body-fitted coordination system.

8.4 Similarity Solution If L is considered as a varying length scale equal to x, then the boundary thickness varies with x as where is the local Reynolds number. A boundary layer flow is similar if its velocity profile as normalized by U depends only on the normalized distance from the wall, , i.e., where V is the velocity components outside the boundary layer normal to U. Here g() and h() are called the similarity variables.

8.4.1 Blasius Solution For uniform flows past a semi-infinite flat plate, the Boundary layer flows are 2-D. It can be shown that the stream function defined by will satisfy the above conditions for similarity solution such that where the f’ denotes the derivative with respect to . Consequently,

8.4.1 Blasius Solution The boundary layer equation in term of the similarity variables becomes: subject to the boundary conditions: The velocity profile obtained by solving the above ordinary differential equation is called the Blasius profile.

8.4.1 Blasius Solution Plot streamwise and transverse velocities

8.4.2 Boundary Thickness and Skin Friction
Since the velocity profile merges smoothly and asymptotically into the free stream, it is difficult to measure the boundary layer thickness . Conventionally,  is defined as the distance from the surface to the point where velocity is 99% of free stream velocity. This occurs when , i.e., Therefore, for laminar boundary layer,

The wall shear stress can be expressed as, And the friction coefficient Cf is given by,

The boundary layer thickness increases with x1/2, while the wall shear stress and the skin friction coefficient vary as x-1/2. These are the characteristics of a laminar boundary layer over a flat plate.

8.5 Turbulent Boundary Layer
Laminar boundary layer flow can become unstable and evolve to turbulent boundary layer flow at down stream. This process is called transition. Among the factor that affect boundary-layer transition are pressure gradient, surface roughness, heat transfer, body forces, and free stream disturbances.

Under typical flow conditions, transition usually occurs at a Reynolds number of 5 x 105, which can be delayed to Re between 3~ 4 x 106 if external disturbances are minimized. Velocity profile of turbulent boundary layer flows is unsteady. Because of turbulent mixing, the mean velocity profile of turbulent boundary layer is more flat near the outer region of the boundary layer than the profile of a laminar boundary layer.

A good approximation to the mean velocity profile for turbulent boundary layer is the empirical 1/7 power-law profile given by This profile doesn't hold in the close proximity of the wall, since at the wall it predicts Hence, we cannot use this profile in the definition of to obtain an expression in terms of .

For the drag of turbulent boundary-layer flow, we use the following empirical expression developed for circular pipe flow, where is the pipe cross-sectional mean velocity and R the pipe radius. For a 1/7-power profile in a pipe, The substitution of and gives,

For turbulent boundary layer, empirically we have Therefore, Experiment shows that this equation predicts the turbulent skin friction on a flat plate within about 3% for 5 x 106 <Rex< 107

Note the friction coefficient for the laminar boundary layer is proportional to Rex-1/2, while that for the turbulent boundary layer is proportional to Rex-1/5, with the proportional constants different also by a factor of 10. The turbulent boundary layer develops more rapidly than the laminar boundary layer.

8.6 Fluid Force on Immersed Bodies
Relative motion between a solid body and the fluid in which the body is immersed leads to a net force, F, acting on the body. This force is due to the action of the fluid. In general, dF acting on the surface element area, will be the added results of pressure and shear forces normal and tangential to the element, respectively.

Hence, The resultant force, F, can be decomposed into parallel and perpendicular components. The component parallel to the direction of motion is called the drag, D, and the component perpendicular to the direction of motion is called the drag, D, and the component perpendicular to the direction of motion is called the lift, L.

Now where is the unit vector inward normal to the body surface, and is the unit vector tangential to the surface along the surface slip velocity direction. The total fluid force on the body becomes

If i is the unit vector in the body motion direction, then magnitude of drag FD becomes: Note that L is in the plane normal to i, generally for three-dimensional flows. For two-dimensional flows, we can denotes j as the unit vector normal to the flow direction.

Therefore, L=FL j where FL is the magnitude of lift and is determined by: For most body shapes of interest, the drag and lift cannot be evaluated analytically Therefore, there are very few cases in which the lift and drag can be determined without resolving by computational or experimental methods.

8.7 Drag The drag force is the component of force on a body acting parallel to the direction of motion. drag force

8.7 Drag The drag coefficient defined as
is a function of Reynolds number , i.e. This form of the equation is valid for incompressible flow over any body, and the length scale, D, depends on the body shape.

8.7.1 Friction Drag If the pressure gradient is zero and no flow separation, then the total drag is equal to the friction drag, , and, The drag coefficient depends on the shear stress distribution.

8.7.1 Friction Drag For a laminar flow over a flat surface, U=U and the skin- friction coefficient is given by, The drag coefficient for flow with free stream velocity, U, over a flat plate of length, L, and width, b, is obtained by substituting into ,

8.7.1 Friction Drag If the boundary layer is turbulent, the shear stress on the flat plate then is given by, The substitution for results in, This result agrees very well with experimental coefficient of, for ReL< 107

8.7.2 Pressure Drag (Form Drag)
The pressure drag is usually associated with flow separation which provide the pressure difference between the front and rear faces of the body. Therefore, this type of pressure drag depends strongly on the shape of the body and is called form drag. In a flow over a flat plate normal to the flow as shown in the following picture, the wall shear stress contributes very little to the drag force. flow separation occurs

The form drag is given by, As the pressure difference between front and rear faces of the plate is caused by the inertia force, the form drag depends only on the shape of the body and is independent of the fluid viscosity.

The drag coefficient for all object with shape edges is essentially independent of Reynolds number. Hence, CD=constant where the constant changes with the body shape and can only be determined experimentally.

8.7.3 Friction and Pressure Drag for Low Reynolds Number Flows
At very low Reynolds number, Re<<1, the viscous force encompass a very large region surrounding the body. The pressure drag is mainly caused by fluid viscosity rather than inertia. Hence, both friction and pressure drags contribute to the total drag force, i.e., the total drag is entirely viscous drag

8.7.3 Friction and Pressure Drag for Low Reynolds Number Flows
For low velocity flows passing a sphere of diameter D, Stokes had shown that the total viscous drag is given by with 1/3 of it being contributed from normal pressure and 2/3 from frictional shear. The drag coefficient then is expressed as As the ReD increases, the flow separates and the relative contribution of viscous pressure drag decreases.

8.8 Drag Coefficient CD for a Sphere as a Function of Re in a Parallel Flow

Chapter 9: FLOWS IN PIPE

9.1 General Concept of Flows in Pipe
As a uniform flow enters a pipe, the velocity at the pipe walls must decrease to zero (no-slip boundary condition). Continuity indicates that the velocity at the center must increase. Thus, the velocity profile is changing continuously from the pipe entrance until it reaches a fully developed condition. This distance, L, is called the entrance length.

For fully developed flows (x>>L), flows become parallel, , the mean pressure remains constant over the pipe cross-section

Flows in a long pipe (far away from pipe entrance and exit region, x>>L) are the limit results of boundary layer flows. There are two types of pipe flows: laminar and turbulent

Whether the flow is laminar or turbulent depends on the Reynolds number, where Um is the cross-sectional mean velocity defined by Transition from laminar to turbulent for flows in circular pipe of diameter D occur at Re=2300

When pipe flow is turbulent. The velocity is unsteadily random (changing randomly with time), the flow is characterized by the mean (time-averaged) velocity defined as: Due to turbulent mixing, the velocity profile of turbulent pipe flow is more uniform then that of laminar flow.

Hence, the mean velocity gradient at the wall for turbulent flow is larger than laminar flow. The wall shear stress, ,is a function of the velocity gradient. The greater the change in with respect to y at the wall, the higher is the wall shear stress. Therefore, the wall shear stress and the frictional losses are higher in turbulent flow.

9.2 Poiseuille Flow Consider the steady, fully developed laminar flow in a straight pipe of circular cross section with constant diameter, D. The coordinate is chosen such that x is along the pipe and y is in the radius direction with the origin at the center of the pipe. y x D b

9.2 Poiseuille Flow For a control volume of a cylinder near the pipe center, the balance of momentum in integral form in x-direction requires that the pressure force, acting on the faces of the cylinder be equal to the shear stress acting on the circumferential area, hence In accordance with the law of friction (Newtonian fluid), have: since u decreases with increasing y

9.2 Poiseuille Flow Therefore: when is constant (negative)
Upon integration: The constant of integration, C, is obtained from the condition of no-slip at the wall. So, u=0 at y=R=D/2, there fore C=R2/4 and finally:

9.2 Poiseuille Flow The velocity distribution is parabolic over the radius, and the maximum velocity on the pipe axis becomes: Therefore, The volume flow rate is:

9.2 Poiseuille Flow The flow rate is proportional to the first power of the pressure gradient and to the fourth power of the radius of the pipe. Define mean velocity as Therefore, This solution occurs in practice as long as, Hence,

9.2 Poiseuille Flow The relation between the negative pressure gradient and the mean velocity of the flow is represented in engineering application by introducing a resistance coefficient of pipe flow, f. This coefficient is a non-dimensional negative pressure gradient using the dynamic head as pressure scale and the pipe diameter as length scale, i.e., Introducing the above expression for (-dp/dx), so,

9.2 Poiseuille Flow At the wall, So,
As a result, the wall friction coefficient is:

9.3 Head Loss in Pipe For flows in pipes, the total energy per unit of mass is given by where the correction factor is defined as, with being the mass flow rate and A is the cross sectional area.

9.3 Head Loss in Pipe So the total head loss between section 1 and 2 of pipes is: hl=head loss due to frictional effects in fully developed flow in constant area conduits hlm=minor losses due to entrances, fittings, area changes, etcs.

9.3 Head Loss in Pipe So, for a fully developed flow through a constant-area pipe, And if y1=y2,

9.3 Head Loss in Pipe For laminar flow, Hence

9.4 Turbulent Pipe Flow For turbulent flows’ we cannot evaluate the pressure drop analytically. We must use experimental data and dimensional analysis. In fully developed turbulent pipe flow, the pressure drop, , due to friction in a horizontal constant-area pipe is know to depend on: Pipe diameter, D Pipe length, L Pipe roughness, e Average flow velocity, Um Fluid density, Fluid viscosity,

9.4 Turbulent Pipe Flow Therefore, Dimensional analysis,
Experiments show that the non-dimensional head loss is directly proportional to L/D, hence

9.4 Turbulent Pipe Flow Defining the friction factor as, , hence
where f is determined experimentally. The experimental result are usually plotted in a chart called Moody Diagram.

9.4 Turbulent Pipe Flow In order to solve the pipe flow problems numerically, a mathematical formulation is required for the friction factor, f, in terms of the Reynolds number and the relative roughness. The most widely used formula for the friction factor is that due to Colebrook, This an implicit equation, so iteration procedure is needed to determine.

9.4 Turbulent Pipe Flow Miller suggested to use for the initial estimate, That produces results within 1% in a single iteration

9.5 Minor Loss The minor head loss may be expressed as,
where the loss coefficient, K, must be determined experimentally for each case. Minor head loss may be expressed as where Le is an equivalent length of straight pipe

9.5 Minor Loss Source of minor loss: 1. Inlets & Outlets
2. Enlargements & Contractions 3. Valves & Fittings 4. Pipe Bends

9.6 Non-Circular Ducts Pipe flow results sometimes can be used for non- circular ducts or open channel flows to estimate the head loss Use Hydraulic Diameter, A - Cross section area; P - Wetted perimeter For a circular duct, For rectangular duct, where Ar =b/a is the geometric aspect ratio

9.6 Non-Circular Ducts Effect of Aspect Ratio (b/a): For square ducts:
For wide rectangular ducts with b>>a: Thus, flows behave like channel flows However, pipe flow results can be used with good accuracy only when: b a a=b Ar=1 Dh=a b a Ar Dh2a b a b a 1/3<Ar<3

Chapter 10: OPEN CHANNEL FLOWS

10.1 General Concept of Flows in Open Channel
Open channel flows are flows with free surface that have many applications Rivers, Streams, Aqueducts, Canals, Sewers, Irrigation Water Channels Pipe Flow vs. Channel Flow Pipe Flow Open Channel Flow Closed with solid boundary Open with free surface Fixed crossed-section Variable depth Driven by pressure gradient Driven by gravity Mostly circular All kind of shapes

Hydraulic radius: Hydraulic depth:

Based on the property of yh, the open channel flow can be classified as: a) Constant depth: 1-D model can be used b) Gradually varying: Depth changes slowly so that 1-D model remains as a good approximation c) Rapidly varying: Need a 2-D model to treat the problem

For most practical cases involving large and deep channels, the Reynolds number, is high Hence, most open channel flows are turbulent With a free surface for open channel flow, gravity is important. The important parameter is the Froude number:

The open channel flows as classified by Froude number Fr are:

10.2 Propagation of Surface Waves
If the free surface initially calm is perturbed by a vertical displacement , there will be an associated velocity perturbation in fluid. If the water depth is small compared with the length scale of the displacement, the displacement perturbation will have a fixed form that propagates with a velocity C, which is called phase velocity. We now consider the flow in a frame moving with the phase velocity. Then, in the moving frame the shape of the displacement is fixed and steady

For small displacement disturbances, Hence, the displacement can propagate upstream and downstream with a speed equal to If the fluid moves at a velocity U, then the Froude number,

This phenomenon can easily be demonstrated by a boat moving at a constant speed U on an initial clam water where the disturbance are generated by the vertical oscillating of the boat. The wave patterns behind the boat for Fr> 1 are called the ship wakes

10.3.1 Friction Loss Bernoulli’s with friction loss,
Since the flow is uniform, For free surface, we also have P1-P2 since the fluid depths are the same. Therefore, where Sb is the hydraulic slope

10.3.1 Friction Loss Similar to pipe flow, so,
Empirical values of C were determined by Manning, who suggested that

Friction Loss Now, For a rectangular channel,

Specific Energy Define specific energy, E, at a single section in the channel as, Let q be volume flow rate per unit width, Q=qb, so for rectangular channel:

Specific Energy The variation of depth as a function of specific energy for a given flow rate are summaries in the specific energy diagram For a given flow rate (Q>0) and specific energy, there are 2 possible values of depth, y. These are called alternate depths. Fr<1 E=y Fr>1

Specific Energy For any value of E, the horizontal distance from the vertical axis to the line, y=E, gives the depth And the distance from the line, y=E, to the Q curve is then equal to the K.E., U2/2g. For each curve representing a given flow rate, there is a value of depth that gives a minimum E. This depth is the critical depth obtained by:

10.3.1 Specific Energy The velocity at the critical condition:
Since , we have Continuity Then

Specific Energy For non-rectangular channels, the channel depth varies across the width. At the minimum specific energy,

Specific Energy Consider,

10.4 Gradually Varies Flow The energy equation for the differential C.V. is: For a rectangular channel U=Q/by, so:

10.4 Gradually Varies Flow Since dz=-sbdx and similarly, we can define dhL= sfdx, the energy equation now becomes, For flow at normal depth, The sign of the slope of the water surface profile depends on whether the flow is subcritical or supercritical, and on sf and sb

10.4 Gradually Varies Flow To calculate the surface profile, rewrite the equation as As dE/dx= sf – sb, and in finite difference form, m denotes the mean properties over a channel length , sf can be obtained from the Manning correlation, since sb for flow at normal depth equal sf

10.5 Hydraulic Jump For subcritical flow, disturbances cause by a change in bed slope or flow cross section may move upstream and downstream. The result is a smooth adjustment of the flow However, when the flow is supercritical, disturbances cannot be transmitted upstream. Thus, a gradual change is not possible. The transition from the supercritical to subcritical flow occurs abruptly through the hydraulic jump

10.5 Hydraulic Jump The abrupt change in depth involves a significant loss of mechanical energy through turbulent mixing The extent of a hydraulic jump is short, so friction is negligible. Assuming horizontal surface, gravitational effect of bottom elevation can be neglected

10.5.1 Depth Change Across a Hydraulic Jump
Eliminate U2 from the momentum equation by using the continuity equation to get:

10.5.2 Head Loss Across a Hydraulic Jump
Head loss through a jump is just the difference in specific energy,

10.6 Flow Over a Bump Consider frictionless flow in a horizontal rectangular channel of constant width, b, with a bump of height, h(x). The flow is assumed uniform. Since the flow is steady, incompressible and frictionless, applying Bernoulli’s equation along the free surface gives:

10.6 Flow Over a Bump Along the free surface, p1=p=patm, thus:

10.6 Flow Over a Bump

Fundamentals of Fluid Mechanics Bruce R. Munson Donald F

Similar presentations

Presentation on theme: "Fundamentals of Fluid Mechanics Bruce R. Munson Donald F"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Fundamentals of Fluid Mechanics Bruce R. Munson Donald F

Similar presentations

Presentation on theme: "Fundamentals of Fluid Mechanics Bruce R. Munson Donald F"— Presentation transcript:

Similar presentations

About project

Feedback