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Pharos University MECH 253 FLUID MECHANICS II

Pharos University MECH 253 FLUID MECHANICS II. Lecture # 5 INVISCID FLOWS. 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  .

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Pharos University MECH 253 FLUID MECHANICS II

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  1. Pharos UniversityMECH 253 FLUID MECHANICS II Lecture # 5 INVISCID FLOWS

  2. 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 Equationswhere , v, and p can be functions of r and t .

  3. 7.1 Inviscid Flow • On the other hand, if flows are steady but compressible, the governing equation becomeswhere  can be a function of r • For compressible flows, the state equation is needed; then, we will require the equation for temperature T also.

  4. 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.

  5. 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

  6. 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. 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.

  8. 7.1 Inviscid Flow • The continuity equation becomeswhich 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 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.

  10. Potential FlowsIntegral Equations • Irrotational Flow • Flow Potential • Conservation of Mass • Laplace Equation

  11. 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

  12. 7.2 2D Potential Flows • The 2-D velocity potential function givesand then the continuity equation becomes • The pressure distribution can be determined by the Bernoulli equation,where p is the dynamic pressure

  13. 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

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

  15. Taking into account: Continuity equation

  16. Irrotational Flow Approximation • For 2D flows, we can also use the stream function • Recall the definition of stream function for planar (x-y) flows • Since vorticity is zero, • This proves that the Laplace equation holds for the stream function and the velocity potential

  17. Cylindrical coordinate system In cylindrical coordinates (r , q ,z ) with -axisymmetric case

  18. Taking into account: Continuity equation

  19. 7.2 Two-Dimensional Potential Flows • 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

  20. 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 .

  21. Irrotational Flow Approximation • Irrotational approximation: vorticity is negligibly small • In general, inviscid regions are also irrotational, but there are situations where inviscid flow are rotational, e.g., solid body rotation

  22. 7.3 Simple 2-D Potential Flows • Uniform Flow • Stagnation Flow • Source (Sink) • Free Vortex

  23. 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

  24. 7.3.1 Uniform Flow • This is a simple uniform flow along a single direction.

  25. Elementary Planar Irrotational FlowsUniform Stream • In Cartesian coordinates • Conversion to cylindrical coordinates can be achieved using the transformation Proof with Mathematica

  26. 7.3.2 Stagnation Flow • For a stagnation flow, . Hence, • Therefore,

  27. y  x 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.

  28. 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,

  29. 7.3.3 Source (Sink) • The integration leads to • Where again the arbitrary integration constants are taken to be zero at . and

  30. 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)

  31. Elementary Planar Irrotational FlowsLine Source/Sink • Potential and streamfunction are derived by observing that volume flow rate across any circle is • This gives velocity components

  32. Elementary Planar Irrotational FlowsLine Source/Sink • Using definition of (Ur, U) • These can be integrated to give  and  Equations are for a source/sink at the origin Proof with Mathematica

  33. 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,

  34. 7.3.4 Free Vortex • The integration leads to where again the arbitrary integration constants are taken to be zero at and

  35. 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.

  36. 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

  37. 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:

  38. 7.4.1 Source and Sink

  39. 7.4.1 Source and Sink • Hence, • Since • We have

  40. 7.4.1 Source and Sink • We have • By • Therefore,

  41. 7.4.1 Source and Sink • The velocity component are:

  42. Elementary Planar Irrotational FlowsDoublet • A doublet is a combination of a line sink and source of equal magnitude • Source • Sink

  43. 7.4.1 Source and Sink

  44. 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

  45. 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

  46. 7.4.2 Doublet • Therefore, as a0 and m with 2am=M • The corresponding velocity components are

  47. 7.4.2 Doublet

  48. 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:

  49. 7.4.3 Source in Uniform Stream

  50. 7.4.3 Source in Uniform Stream • 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

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