1 MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 7: INVISCID FLOWS Instructor: Professor C. T. HSU
2 7.1 Inviscid FlowInviscid 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 .
3 7.1 Inviscid FlowOn the other hand, if flows are steady but compressible, the governing equation becomes where can be a function of rFor compressible flows, the state equation is needed; then, we will require the equation for temperature T also.
4 7.1 Inviscid FlowCompressible 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.
5 7.1 Inviscid FlowFor steady incompressible flow, the governing eqt reduce further to where = constant.The equation of motion can be rewrited intoTake the scalar products with dr and integrate from a reference at along an arbitrary streamline =C , leads to since
6 7.1 Inviscid FlowIf 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 areSince is the vorticity , flows with are called irrotational flows.
7 7.1 Inviscid FlowNote 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.
8 7.1 Inviscid FlowThe 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.
9 7.1 Inviscid FlowThe 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 haveHowever, the pressure cannot be superposed due to the nonlinearity in the Bernoulli equation, i.e.
10 7.2 2D Potential FlowsIf restricted to steady two dimensional potential flow, then the governing equations becomeE.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 /z0.ULyxzD
11 7.2 2D Potential FlowsThe 2-D velocity potential function gives and then the continuity equation becomesThe pressure distribution can be determined by the Bernoulli equation, where p is the dynamic pressure
12 7.2 2D Potential FlowsFor 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
13 7.2 Two-Dimensional Potential Flows For two-dimensional flows, become:In a Cartesian coordinate systemIn a Cylindrical coordinate systemandand
14 7.2 Two-Dimensional Potential Flows Therefore, there exists a stream function such thatin the Cartesian coordinate system andin the cylindrical coordinate system.The transformation between the two coordinate systems
15 7.2 Two-Dimensional Potential Flows 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
17 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.andand
18 7.3.1 Uniform FlowThis is a simple uniform flow along a single direction.
19 7.3.2 Stagnation FlowFor a stagnation flow, Hence,Therefore,
20 7.3.2 Stagnation FlowThe flow an incoming far field flow which is perpendicular to the wall, and then turn its direction near the wallThe origin is the stagnation point of the flow. The velocity is zero there.xy
21 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,
22 7.3.3 Source (Sink) The integration leads to and Where again the arbitrary integration constants are taken to be zero atand
23 7.3.3 Source (Sink)A pure radial flow either away from source or into a sinkA +ve m indicates a source, and –ve m indicates a sinkThe magnitude of the flow decrease as 1/rz direction = into the paper. (change graphics)
24 7.3.4 Free VortexConsider 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,
25 7.3.4 Free VortexThe integration leads towhere again the arbitrary integration constants are taken to be zero atand
26 7.3.4 Free VortexThe potential represents a flow swirling around origin with a constant circulation .The magnitude of the flow decrease as 1/r.
27 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 SinkDoubletSource in Uniform Stream2-D Rankine OvalsFlows Around a Circular Cylinder
28 7.4.1 Source and SinkConsider 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 isThen the stream function and potential function obtained by superposition are given by:
34 7.4.2 DoubletThe 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
35 7.4.2 Doublet For source of m at (-a,0) and sink of m at (a,0) Under these limiting conditions of a0, m , we have
36 7.4.2 Doublet Therefore, as a0 and m with 2am=M The corresponding velocity components are
40 7.4.3 Source in Uniform Stream The velocity components are:A stagnation point occurs atTherefore, the streamline passing through thestagnation point whenThe maximum height of the curve is
41 7.4.3 Source in Uniform Stream 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.
42 D Rankine OvalsThe 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,
47 7.4.5 Flows Around a Circular Cylinder Steady CylinderRotating CylinderLift Force
48 Steady CylinderFlow around a steady circular cylinder is the limiting case of a Rankine oval when a0.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.
49 Steady CylinderThe stream function and velocity potential become:The radial and circumferential velocities are:
51 Rotating CylinderThe 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
52 Rotating CylinderThe radial and circumferential velocities are given by
61 Lift ForceThe 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,
62 Lift ForceThe surface pressure as obtained from Bernoulli equation iswhere is the pressure at far away from the cylinder.
63 Lift ForceHence,The force due to pressure in x and y directions are then obtained by
64 Lift ForceThe 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.