Chapter 3: Fundamentals of Inviscid, Incompressible Flow

1. Chapter 3: Fundamentals of Inviscid, Incompressible Flow SONG, Jianyu Mar. 8th.2009

2. Part 2, Inviscid, Incompressible Flow • Inviscid: frictionless or perfect fluid • Incompressible: the density is constant

3. What will we learn from this chapter? • Bernoulli’s equation • Laplace’s equation • Some elementary flows

4. Bernoulli’s equation

5. Bernoulli’s equation Recall the x component of the momentum equation: The form of substantial derivative: Translate the equation into substantial derivative form: ------------------------------------------- For inviscid flow with no body forces: For steady flow: L.H.S The Math Fact:

6. Euler’s equation • Inviscid flow with no body force • It relate the change in velocity along a streamline dV to the change in pressure dp along the same steam line

7. Bernoulli’s equation For incompressible flow: ρ=constant Integrated between any two point along a streamline Bernoulli’s equation holds along a streamline in either rotational or irrotational case. However, for rotational flow it will change from different streamline. If the flow is irrotational, it is constant for the flow (Problem 1) Bernoulli’s equation

8. Laplace’s equation

9. Stream Function Consider 2D steady flow Equation for a streamline is It can be integrated to yield the algebraic equation for a streamline Now use symbol instead of f The function is called stream function. To get a streamline we only have to set c be some constant. Figure 2.40 Let us define the numerical value of such that the difference between two streamlines is equal to the mass flow between the two streamlines. Since it is 2D flow, the mass flow between two streamlines is defined per unit depth perpendicular to the page Thus choose one streamline of the flow, and give it an arbitrary value of the stream function. The value of the stream function for any other streamline in the flow is fixed.

10. Stream Function Let be the normal distance between “ab” and “cd”. Choose it be small enough so that V is constant across Figure 2.41: Due to conservation of mass According to the chain rule: For incompressible flow:

11. Velocity Potential For an irrotational flow: Consider the following vector identify: If φ is a scalar function then (Due to the trivial Math fact: The curl of the gradient of a scalar function is identically zero. Comparing equations and we see that) For an irrotational flow, there exists a scalar function φ such that the velocity is given by the gradient of φ. We denote φ as velocity potential. Since the definition of gradient in Cartesian coordinate is: Thus: By the way, in cylindrical coordinate, it is:

12. CMP Stream function Velocity potential φ • By differentiating normal to the velocity direction • Either irrotational or rotational flow • Only Define to 2D flow • Flow field velocities by differentiating in the same direction • Only for irrotational flow • Apply to 3D

13. CMP We see that a line of constant φ is an isoline of φ; Since φ is the velocity potential we give this isoline a specific name equipotential line. since This gradient line is a streamline See P171~172 for the detail proof of the streamline and the equipotential line are perperndicular to each other.

14. Laplace’s equation Recall the continuity equation: For incompressible flow ρ=constant For irrotational flow, it has velocity potential function: Solutions of Laplace’s equation are called harmonic functions. ------------------------------------------ For 2D incompressible flow, stream function can be defined In fact, it automatically satisfied the condition. For irrotational flow: Laplace’s equation Laplace’s equation

15. Laplace’s equation • Any irrotational, incompressible flow has a velocity potential and stream function(for 2D) that both satisfy Laplace’s equation. • Conversely, any solution of Laplace’s equation represents the velocity potential or stream function (2D) for an irrotational incompressible flow • Note that Laplace’s equation is linear , the sum of any particular solution of a linear differential equation is also a solution of the equation, so a complicated flow pattern for an irrotational incompressible flow can be synthesize by adding together a number of elemental flows that are also irrotational and incompressible.

16. Boundary Conditions Different flows for the different bodies are all governed by the Laplace equation, but the boundary conditions are different. Infinity Boundary Conditions Wall Boundary Conditions For inviscid flows the velocity at the surface can be finite, but because the flow cannot penetrate the surface, the velocity vector must be tangent to the surface. Express in velocity potential: Express in streamline function: Where “s” is the distance measured along the body surface The body shape function is given as: The shape of the body surface can be expressed as: Express directly in u and v:

17. Some elementary flows

18. Uniform Flow It is easy to show that the uniform flow is incompressible and irrotational since it satisfies : Hence we can use velocity potential: For x axils and y axils: Integrating them w.r.t x and y respectively Compare them we get: The actual value of the velocity potential is not important. So without loss of rigor, we get : For incompressible flow, we can express it in the form of streamline function: In a similar way, we get:

19. Source Flow Let the velocity along each of the streamlines vary inversely with distance for O. The opposite case is the sink flow which is simply a negative source flow It is easy to show that At every pint except the origin where it becomes infinite so the origin is a singular point. And it is irrotational every where since Hence:

20. Source Flow The total mass flow across the surface of the cylinder is : The rate of volume is : Define the volume flow rate per unit length along the cylinder as source strength For which a positive one represents the source while the negative one represents the sink.

21. Source Flow Express the flow in velocity potential: Integrating them w.r.t. r and θ respectively Compare and get ride of the constant : For the stream function: Integrating them w.r.t. r and θ respectively Compare and get ride of the constant :

22. Doublet Flow This is a special, degenerate case of a source-sink pair that leads to a singularity called a doublet. At any point P, the stream function is: Let L->0 while lΛ remains constant Denote the strength of the doublet is denoted by The streamline function: In a similar way, we can get the velocity potential:

23. Vortex Flow Streamlines are concentric circles about a given point. The velocity along any given circular streamline be constant, but let it vary from one streamline to another inversely with distance from the common center. (1)Vertex flow is physically possible incompressible flow, that is at every point: (2)Vortex flow is irrotational , that is at every point except the origin. The velocity potential can be get in the following: The stream function can be get in a similar way:

24. Vortex Flow What is the value at r=0? Since And dS has the same direction Let r->0 ,and the circulation will still remain But r->0 so ds->0

25. Thank you! Q&A