Advance Fluid Mechanics Course Number: 804538 Advance Fluid Mechanics Faculty Name Prof. A. A. Saati

804538 Advanced Fluid Mechanics Course Number: 804538 Course Title: Advanced Fluid Mechanics Credit ( Lec, Lab, Cr ): (3, 0, 3) Prerequisite: 804305 or A. C. Course Objective: This course gives the students insight into the phenomena of viscous fluid flow, to enable them derive the governing equations for practical cases and to show how the boundary layer theory can make flows involving fluids of small viscosity amenable to successful theoretical analysis

804538 Advanced Fluid Mechanics Course Outline: Conservation equations for viscous fluids – boundary layer concept – Navier-Stokes equation and some exact solutions – Laminar boundary layer equations and methods of solution Von Karman momentum integral equation – Theory of stability of laminar flows – Introduction to turbulent flow.

804538 Advanced Fluid Mechanics

Ref. ADVANCED FLUID MECHANICS By W. P. Graebel Lecture # 1 Chapter: 1 Fundamentals Ref. ADVANCED FLUID MECHANICS By W. P. Graebel

1.1 Introduction The fundamentals laws of fluid mechanics: conservation of mass Newton's laws, and the laws of thermodynamics

1.1 Introduction The quantities of mass, momentum and energy for a given volume of fluid mechanics will change due to: internal causes net change in that quantity entering and leaving action on the surface

In some cases a global description (by applying the basic laws) is satisfactory for carrying out further analysis, but A local statement of the laws in the form of differential equations is preferred to obtain more detailed information on the behavior of the quantity.

Divergence theorem is a way to convert certain types of surface integrals to volume integrals Where: The theorem assumes that the scalar and vector quantities are finite and continuous within V and on S.

In studying fluid mechanic three laws can be expressed in the following descriptive form:

The general term used to classify the fluid mechanic quantities is tensor. The order of a tensor refers to the number of directions associated with the term: Example: pressure and temperature, have magnitude but zero directional property Velocity, have magnitude but one direction associated with them Stress, have magnitude but two directions associated with them

To qualify as a tensor, quantity must have more than the magnitude and directionality. When the components of the tensor are compered in two coordinate systems that have origins at the same point, the components must relate one another in a specific manner. Example: Tensor of order zero, the transformation law is simply that the magnitude are the same in both coordinate systems Tensor of order one, must transform according to the parallelogram law (the component in one coordinate system are the sum of products of direction cosines of the angles between the two sets of axes and the components in the second system).

Cartesian coordinates When dealing with flows that involve flat surfaces. Boundary conditions are most easily satisfied, manipulation are easiest,…… The components of a vector are represented by:

1.2 Velocity, Acceleration, and the Material Derivative A fluid is defined as material that will undergo constant motion when shearing forces are applied, the motion continuing as long as the sheering forces are maintained.

1.2 Velocity, Acceleration, and the Material Derivative Velocity of the particle is given by:

1.2 Velocity, Acceleration, and the Material Derivative Acceleration of the particle is given by: Notes: that the are fixed for the partial derivatives

1.2 Velocity, Acceleration, and the Material Derivative Eularian or spatial description The velocity is written as , Where x refers to the position of a fixed point in space, as the basic descriptor rather than displacement. The acceleration component in the x direction is defined as:

1.2 Velocity, Acceleration, and the Material Derivative Eularian or spatial description Similar results for acceleration component in the y and z direction is defined as: And the vector form is defined as:

1.2 Velocity, Acceleration, and the Material Derivative Eularian or spatial description The first term is referred to the temporal acceleration, and the second as convective or advective, acceleration. The convective term can also be written as

1.2 Velocity, Acceleration, and the Material Derivative Eularian or spatial description Material or substantial derivative represents differentiation as fluid partial is monitored Note that is not correct vector operator.

1.3 The Local Continuity Equation A Control volume is a device used in analyzing fluid flows to account for mass, momentum, and energy balance A control surface is the bounding surface of the control volume. The mass of the fluid inside our control volume is The rate of change of mass inside of our control volume is (for fixed VC in space) The rate of change of mass inside of our control volume through its surface dS is

1.3 The Local Continuity Equation The net rate of change of mass inside and entering the control volume and setting the sum to zero is given by Transform the surface integral to a volume integral Making this replacement in the above equation

1.3 The Local Continuity Equation Since the choice of the control volume was arbitrary and since the integral must vanish no matter what choice of control volume was made, the only way this integral can vanish is for the integrand to vanish

1.3 The Local Continuity Equation An incompressible is defined as one where the mass density of the fluid particle does not change as the particle followed

1.4 Path Line, Streamline and Stream Function A path line is a line along which a fluid particle actually travels. Since the particle incrementally moves in the direction of the velocity vector, the equation of a path line is given by Note : the integration being performed with held fixed A streamline is defined as line drawn in the flow at a given instant of time such that the fluid velocity vector at any point in the streamline is tangent to the line at that point.

1.4 Path Line, Streamline and Stream Function A streamline is defined as line drawn in the flow at a given instant of time such that the fluid velocity vector at any point in the streamline is tangent to the line at that point. The streamline are given by equation A stream surface is a collection of adjacent streamlines, providing a surface through which there is no flow A stream tube is a tube made up of adjoining streamlines

1.4 Path Line, Streamline and Stream Function For steady flows the path lines and streamlines overlap For unsteady flows the path lines and streamlines may differ Stream functions are used mainly in connection with incompressible flows (where the density of individual particle does not change) and the equation reduce to

1.4.1 Lagrange’s Stream Function for Tow-Dimensional Flows For 2-D flows equation reduces to Integrating that one of the velocity components can be expressed in tem of the other. Lagrange’s stream function is a scalar function , and expressing the two velocity in terms of .

1.4.1 Lagrange’s Stream Function for Tow-Dimensional Flows In 2-D the tangency requirement equation Using the expression of velocity in term of stream function This equation states that along a streamline vanishes, and is constant. The suitable expressions for the velocity component in cylindrical polar coordinates are

1.4.2 Stream Function for Three-Dimensional Flows For 3-D equation states that there is one relationship between the three velocity components So it is expected that the velocity can be expressed in terms of two scalar function The clarification of stream function as introduced in two dimensions is Where and are each constant on stream surfaces.

1.4.2 Stream Function for Three-Dimensional Flows A particular three dimensional case in which a stream function is useful is that of axisymmetric flow. Taking the z-axis as the axis of symmetry: Spherical polar coordinates Cylindrical polar coordinates

1.4.2 Stream Function for Three-Dimensional Flows Since any plane given by equal to constant is therefore a stream function given by Spherical polar coordinates Cylindrical polar coordinates

1.5 Newton’s Momentum Equation The momentum in the interior of the control volume is The rate at which momentum enters the control volume through its surface is The net rate of change of momentum is then

1.5 Newton’s Momentum Equation The forces applied to the surface of the control volume are due to Pressure force Viscous force Gravity force The net force is then The net change in momentum gives Using the divergence theorem, this reduce to

1.6 Stress Stress is defined as force applied to an area Consider the three special stress vectors corresponding to forces acting on areas unit normal pointing in the x, y, z, directions (n=i, n=j, n=k)

1.6 Stress In the limit, as the areas are taken smaller and smaller, the forces acting are Where acting on -x, -y, -z, directions And The summing result is In limiting the areas goes to zero

1.6 Stress Combine the above equations (*) & (**) Integrate the above equation and change the surface integral to volume integral

1.6 Stress Integrate the above equation and change the surface integral to volume integral

1.6 Stress Integrate the above equation The net change in momentum equation gives As was the case for continuity equation, the above equation must be valid no matter what volume we choose Therefore, it must be that

Stress vector Stress tensor

1.6 Stress The continuity equation and The momentum equation in (11) can be written in a matrix form as This form referred to as conservation form and frequently used in computational fluid dynamic

1.6 Stress Moments can be balanced in the same manner as forces Using a finite control mass and taking R as a position vector drawn from the point about which moments are being taken From equations (2) &(4) {ppt 36}; The time rate of change of moment of momentum is given by

1.6 Stress And using the product rule Also using the product rule to develop The surface integral

1.6 Stress Thus, using equations (14, 15, 16, 17) in equation (13)

END OF LECTURE 1

1.7 Rates of Deformation Rate of deformation is the quantity that describes the fluids’ behavior under stress. Select points A,B,C, to make up a right angle at an initial time t At later time , these points will have moved to A’,B’, and C’ Point A will have x and y velocity components

1.7 Rates of Deformation Point A will have x and y velocity components And will move a distance Similarly for point B&C to B’&C’

1.7 Rates of Deformation Normal rates of deformation: The rate of change of length along the axis per unit length, which will be denoted by

1.7 Rates of Deformation The rate of change of volume per unit volume is the volume at minus the original volume at t divided by the original volume times , This is the dilatational strain rate

1.7 Rates of Deformation Besides changes of length, changes of angles are involved in the deformation. The angles that AB and BC have rotated through And similarly

1.7 Rates of Deformation The sum of , represent the rate of change of the angle ABC. Let be defined as rat of shear deformation

1.7 Rates of Deformation rates of shear deformation: The rate of change of the angle ABC in each plane is

1.7 Rates of Deformation Consider any two neighboring fluid particles a distance dr apart, The distance dr changes with time but must remain small because the particles were initially close together To find the rate at which the particles separate, Take the time derivative of dr Where dv is the difference in velocity between the two particles

1.7 Rates of Deformation Since the magnitude of the distance between the two points is

1.7 Rates of Deformation To find the rate at which the distance between two points change, we need to know

1.8 constitutive of Relations Considering a fluid of simple molecular structure, such as water or air, experience and many experiments suggest the following: 1. Stress will depend explicitly only on pressure and the rate of deformation. Temperature can enter only implicitly through coefficients such as viscosity. 2. When the rate of deformation is identically zero, all shear stresses vanish, and the normal stresses are each equal to the negative of the pressure. 3. The fluid is isotropic. That is, the material properties of a fluid at any given point are the same in all directions. 4. The stress must depend on rate of deformation in a linear manner, according to the original concepts of Newton.

1.8 constitutive of Relations The most general relation is

1.8 constitutive of Relations The definition of pressure varies in different instances. In elementary thermodynamics texts, the term pressure is commonly used for the negative of mean normal stress

1.8 constitutive of Relations A state equation is necessary addition to the constitutive description of the fluids. Examples

1.9 Equations for Newtonian Fluid (Navier-Stokes equation) Using equations in section 1.8 and 1.6 we have

(Navier-Stokes equation) 1.9 Equations for Newtonian Fluid (Navier-Stokes equation) When and are constant for incompressible And the continuity equation is the vector for of Newtonian equation given by

(Navier-Stokes equation) 1.9 Equations for Newtonian Fluid (Navier-Stokes equation) The Navier-Stokes equation can be written in component form as

END OF LECTURE 2

1.11 Vorticity and Circulation Any motion of a small region of a fluid can be thought of as combination of translation, rotation and deformation Translation is described by velocity of the point Deformation is described by rates of deformation Hear we consider the rotation of a fluid element In considering the transformation of ABC into A’B’C’ , This show that, the angle ABC has changed or deformed by and bisector of the angle ABC has rotated by

1.11 Vorticity and Circulation the angle ABC has rotated by (See sec. 1.7) The rotation of a fluid element is seen to be (a) The curl of v in Cartesian coordinates (b) From equations (a) and (b) show that the equation (a) is one half of the z component (in xy – plane)

1.11 Vorticity and Circulation Similar arguments in the yz and xz planes would yield to x and y components The rotation of a fluid element is seen to be one half the curl of the velocity Therefore the vorticity vector is define as The definition agrees with the usual “right-hand-rule” sign convention of angular rotation Vorticity also can be represented as a second-order tensor

1.11 Vorticity and Circulation The rate of deformation and vorticity can be represented in index notation as follows: ratational flows: flows with vorticity irratational flows: flows without vorticity vortex lines : is defined as a lines instantaneously tangent to the vorticity vector, satisfying the equations

1.11 Vorticity and Circulation vortex sheets: are surfaces of vorticity lines lying side by side. vortex tubes: are closed vortex sheets with vorticity entering and leaving through the end of the tube The concept of volume flow through an area circulation : is defined as the vorticity flow through an area. Where: c is closed path bounding the area S note: the relation between the line and surface integral is follow form Stokes theorem

1.12 The Vorticity Equation Differential equations governing the change of vorticity can be formed from Navier-Stokes equation Dividing the above equation by the mass density and then taking the curl of the equation and the use of continuity equation, the result is The Vorticity Equation

1.12 The Vorticity Equation The vorticity can be change by three mechanisms: The first term is the vorticity change due to vortex line stretching The second term is the vorticity change by the density gradient, (that when the pressure and density gradient are parallel) The third term the vorticity will diffused by viscosity

1.12 The Vorticity Equation The circulation taken over any cross-sectional area of a vortex tube is a constant On the vorticity in normal to the surface, so the integral is zero. or

1.12 The Vorticity Equation Vortex lines can neither originate nor terminate in the interior of the flow. Either they are closed curves (e.g. smoke rings) or they originate at the boundary. The rate of circulation Since the second term in the integral is around a closed path give zero, then the rate of change of circulation is given by

1.13 The Work-Energy Equation Another useful equation derived from Navier-Stokes is a work-Energy statement.

1.13 The Work-Energy Equation Another useful equation derived from Navier-Stokes is a work-Energy statement.

1.13 The Work-Energy Equation The function represents the rate of dissipation of energy viscosity ( dissipation function )

1.14 The First Law of Thermodynamics The conservation of energy: it is the rate of change of energy of the system is equal to the rate of heat addition to the system plus the rate at which work is done on the system The rate of energy change Where e specific energy and u specific internal energy

1.14 The First Law of Thermodynamics The rate of heat added to the system Where q is heat flux vector, and dr/dt is the heat generated either internally or transferred by radiation The rate of work is being done on the system

1.14 The First Law of Thermodynamics Putting the above equations into the general equation then we have With the help of divergence theorem

1.14 The First Law of Thermodynamics With arbitrary control volume

1.15 Dimensionless Parameters The most commune came from the Navier-Stokes equation and their boundary conditions. Reynolds number Froude number Richardson number Strouhal number Pressure coefficient Drag coefficient Lift coefficient Moment coefficient Weber number

END OF CHAPTER 1 H.W. 1.2, 1.5, 1.7, 1.9