Chapter 9: Differential Analysis of Fluid Flow

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1 of Fluid Flow

2 Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known velocity field. 3. Obtain analytical solutions of the equations of motion for simple flows. Fondamenti di Meccanica dei Continui 2

3 Introduction Recall Chap 5: Control volume (CV) versions of the laws of conservation of mass and energy Chap 6: CV version of the conservation of momentum CV, or integral, forms of equations are useful for determining overall effects However, we cannot obtain detailed knowledge about the flow field inside the CV motivation for differential analysis Fondamenti di Meccanica dei Continui 3

4 Introduction Example: incompressible Navier-Stokes equations We will learn: Physical meaning of each term How to derive How to solve Fondamenti di Meccanica dei Continui 4

5 Introduction For example, how to solve? Step Analytical Fluid Dynamics (Chapter 9) Computational Fluid Dynamics (Chapter 15) Setup Problem and geometry, identify all dimensions and parameters List all assumptions, approximations, simplifications, boundary conditions Simplify PDE s Build grid / discretize PDE s Integrate equations Apply I.C. s and B.C. s to solve for constants of integration Verify and plot results Solve algebraic system of equations including I.C. s and B.C s Verify and plot results Fondamenti di Meccanica dei Continui 5

6 Conservation of Mass Recall CV form (Chap 5) from Reynolds Transport Theorem (RTT) We ll examine two methods to derive differential form of conservation of mass Divergence (Gauss) Theorem Differential CV and Taylor series expansions Fondamenti di Meccanica dei Continui 6

7 Conservation of Mass Divergence Theorem Divergence theorem allows us to transform a volume integral of the divergence of a vector into an area integral over the surface that defines the volume. Fondamenti di Meccanica dei Continui 7

8 Conservation of Mass Divergence Theorem Rewrite conservation of mass Using divergence theorem, replace area integral with volume integral and collect terms Integral holds for ANY CV, therefore: Fondamenti di Meccanica dei Continui 8

9 Conservation of Mass Differential CV and Taylor series First, define an infinitesimal control volume dx x dy x dz Next, we approximate the mass flow rate into or out of each of the 6 faces using Taylor series expansions around the center point, e.g., at the right face Ignore terms higher than order dx Fondamenti di Meccanica dei Continui 9

10 Conservation of Mass Differential CV and Taylor series Infinitesimal control volume of dimensions dx, dy, dz Area of right face = dy dz Mass flow rate through the right face of the control volume Fondamenti di Meccanica dei Continui 10

11 Conservation of Mass Differential CV and Taylor series Now, sum up the mass flow rates into and out of the 6 faces of the CV Net mass flow rate into CV: Net mass flow rate out of CV: Plug into integral conservation of mass equation Fondamenti di Meccanica dei Continui 11

12 Conservation of Mass Differential CV and Taylor series After substitution, y z Dividing through by volume dxdydz y z Or, if we apply the definition of the divergence of a vector Fondamenti di Meccanica dei Continui 12

13 Conservation of Mass Alternative form Use product rule on divergence term Fondamenti di Meccanica dei Continui 13

14 Conservation of Mass Cylindrical coordinates There are many problems which are simpler to solve if the equations are written in cylindrical-polar coordinates Easiest way to convert from Cartesian is to use vector form and definition of divergence operator in cylindrical coordinates Fondamenti di Meccanica dei Continui 14

15 Conservation of Mass Cylindrical coordinates Fondamenti di Meccanica dei Continui 15

16 Conservation of Mass Special Cases Steady compressible flow Cartesian y z Cylindrical Fondamenti di Meccanica dei Continui 16

17 Conservation of Mass Special Cases Incompressible flow and ρ = constant Cartesian y z Cylindrical Fondamenti di Meccanica dei Continui 17

18 Conservation of Mass In general, continuity equation cannot be used by itself to solve for flow field, however it can be used to 1. Determine if velocity field is incompressible 2. Find missing velocity component Fondamenti di Meccanica dei Continui 18

19 The Stream Function Consider the continuity equation for an incompressible 2D flow Substituting the clever transformation y Gives This is true for any smooth function ψ(x,y) Fondamenti di Meccanica dei Continui 19

20 The Stream Function Why do this? Single variable ψ replaces (u,v). Once ψ is known, (u,v) can be computed. Physical significance 1. Curves of constant ψ are streamlines of the flow 2. Difference in ψ between streamlines is equal to volume flow rate between streamlines Fondamenti di Meccanica dei Continui 20

21 The Stream Function Physical Significance Recall from Chap. 4 that along a streamline Change in ψ along streamline is zero Fondamenti di Meccanica dei Continui 21

22 The Stream Function Physical Significance Difference in ψ between streamlines is equal to volume flow rate between streamlines Fondamenti di Meccanica dei Continui 22

23 Conservation of Linear Momentum Recall CV form from Chap. 6 Body Force Surface Force σ ij = stress tensor Using the divergence theorem to convert area integrals Fondamenti di Meccanica dei Continui 23

24 Conservation of Linear Momentum Substituting volume integrals gives, Recognizing that this holds for any CV, the integral may be dropped This is Cauchy s Equation Can also be derived using infinitesimal CV and Newton s 2nd Law (see text) Fondamenti di Meccanica dei Continui 24

25 Conservation of Linear Momentum Alternate form of the Cauchy Equation can be derived by introducing (Chain Rule) Inserting these into Cauchy Equation and rearranging gives Fondamenti di Meccanica dei Continui 25

26 Conservation of Linear Momentum Unfortunately, this equation is not very useful 10 unknowns Stress tensor, σ ij : 6 independent components Density ρ Velocity, V : 3 independent components 4 equations (continuity + momentum) 6 more equations required to close problem! Fondamenti di Meccanica dei Continui 26

27 σ ij = Navier-Stokes Equation First step is to separate σ ij into pressure and viscous stresses σ xx σ xy σ xz p 0 0 τ xx τ xy τ xz σ yx σ yy σ yz = 0 p 0 σ zx σ zy σ + τ yx τ yy τ yz zz 0 0 p τ zx τ zy τ zz Situation not yet improved 6 unknowns in σ ij 6 unknowns in τ ij + 1 in P, which means that we ve added 1! Viscous (Deviatoric) Stress Tensor Fondamenti di Meccanica dei Continui 27

28 Navier-Stokes Equation (paint) (toothpaste) (quicksand) Reduction in the number of variables is achieved by relating shear stress to strainrate tensor. For Newtonian fluid with constant properties Newtonian fluid includes most common fluids: air, other gases, water, gasoline Fondamenti di Meccanica dei Continui 28 Newtonian closure is analogous to Hooke s law for elastic solids

29 Navier-Stokes Equation Substituting Newtonian closure into stress tensor gives = _ Using the definition of ε ij (Chapter 4) Fondamenti di Meccanica dei Continui 29

30 Navier-Stokes Equation Substituting σ ij into Cauchy s equation gives the Navier-Stokes equations Incompressible NSE written in vector form This results in a closed system of equations! 4 equations (continuity and momentum equations) 4 unknowns (U, V, W, p) Fondamenti di Meccanica dei Continui 30

31 Navier-Stokes Equation In addition to vector form, incompressible N-S equation can be written in several other forms Cartesian coordinates Cylindrical coordinates Tensor notation Fondamenti di Meccanica dei Continui 31

32 Navier-Stokes Equation Cartesian Coordinates Continuity X-momentum Y-momentum Z-momentum See page 431 for equations in cylindrical coordinates Fondamenti di Meccanica dei Continui 32

33 Navier-Stokes Equation Tensor and Vector Notation Tensor and Vector notation offer a more compact form of the equations. Tensor notation Continuity Vector notation Tensor notation Conservation of Momentum Vector notation Repeated indices are summed over j (x 1 =x, x 2 = y, x 3 = z, U 1 = U, U 2 = V, U 3 = W) Fondamenti di Meccanica dei Continui 33

34 Differential Analysis of Fluid Flow Problems Now that we have a set of governing partial differential equations, there are 2 problems we can solve 1. Calculate pressure (P) for a known velocity field 2. Calculate velocity (U, V, W) and pressure (P) for known geometry, boundary conditions (BC), and initial conditions (IC) Fondamenti di Meccanica dei Continui 34

35 Exact Solutions of the NSE There are about 80 known exact solutions to the NSE The can be classified as: Linear solutions where the convective term is zero Nonlinear solutions where convective term is not zero Solutions can also be classified by type or geometry 1. Couette shear flows 2. Steady duct/pipe flows 3. Unsteady duct/pipe flows 4. Flows with moving boundaries 5. Similarity solutions 6. Asymptotic suction flows 7. Wind-driven Ekman flows Fondamenti di Meccanica dei Continui 35

36 Exact Solutions of the NSE Procedure for solving continuity and NSE 1.Set up the problem and geometry, identifying all relevant dimensions and parameters 2.List all appropriate assumptions, approximations, simplifications, and boundary conditions 3.Simplify the differential equations as much as possible 4.Integrate the equations 5.Apply BC to solve for constants of integration 6.Verify results Fondamenti di Meccanica dei Continui 36

37 Boundary conditions Boundary conditions are critical to exact, approximate, and computational solutions. Discussed in Chapters 9 & 15 BC s used in analytical solutions are discussed here No-slip boundary condition Interface boundary condition These are used in CFD as well, plus there are some BC s which arise due to specific issues in CFD modeling. These will be presented in Chap. 15. Inflow and outflow boundary conditions Symmetry and periodic boundary conditions Fondamenti di Meccanica dei Continui 37

38 No-slip boundary condition For a fluid in contact with a solid wall, the velocity of the fluid must equal that of the wall Fondamenti di Meccanica dei Continui 38

39 Interface boundary condition When two fluids meet at an interface, the velocity and shear stress must be the same on both sides The latter expresses the fact that when the interface is in equilibrium, the sum of the forces over it is zero. If surface tension effects are negligible and the surface is nearly flat: Fondamenti di Meccanica dei Continui 39

40 Interface boundary condition Degenerate case of the interface BC occurs at the free surface of a liquid. Same conditions hold Since µ air << µ water, As with general interfaces, if surface tension effects are negligible and the surface is nearly flat P water = P air Fondamenti di Meccanica dei Continui 40

41 Example exact solution (Ex. 9-15) Fully Developed Couette Flow For the given geometry and BC s, calculate the velocity and pressure fields, and estimate the shear force per unit area acting on the bottom plate Step 1: Geometry, dimensions, and properties Fondamenti di Meccanica dei Continui 41

42 Example exact solution (Ex. 9-15) Fully Developed Couette Flow Step 2: Assumptions and BC s Assumptions 1. Plates are infinite in x and z 2. Flow is steady, / t = 0 3. Parallel flow, the vertical component of velocity v = 0 4. Incompressible, Newtonian, laminar, constant properties 5. No pressure gradient 6. 2D, w=0, / z = 0 7. Gravity acts in the -z direction, Boundary conditions 1. Bottom plate (y=0) : u = 0, v = 0, w = 0 2. Top plate (y=h) : u =V, v = 0, w = 0 Fondamenti di Meccanica dei Continui 42

43 Example exact solution (Ex. 9-15) Fully Developed Couette Flow Step 3: Simplify 3 6 Note: these numbers refer to the assumptions on the previous slide Continuity X-momentum This means the flow is fully developed or not changing in the direction of flow 2 Cont Cont. 6 Fondamenti di Meccanica dei Continui 43

44 Example exact solution (Ex. 9-15) Fully Developed Couette Flow Step 3: Simplify, cont. Y-momentum 2, , Z-momentum 2, Fondamenti di Meccanica dei Continui 44

45 Example exact solution (Ex. 9-15) Fully Developed Couette Flow Step 4: Integrate X-momentum integrate integrate Z-momentum integrate (in fact the constant C 3 should - in general be a function of y and z ) Fondamenti di Meccanica dei Continui 45

46 Example exact solution (Ex. 9-15) Fully Developed Couette Flow Step 5: Apply BC s y=0, u=0=c 1 (0) + C 2 C 2 = 0 y=h, u=v=c 1 h C 1 = V/h This gives For pressure, no explicit BC, therefore C 3 can remain an arbitrary constant (recall only P appears in NSE). Let p = p 0 at z = 0 (C 3 renamed p 0 ) 1. Hydrostatic pressure 2. Pressure acts independently of flow Fondamenti di Meccanica dei Continui 46

47 Example exact solution (Ex. 9-15) Fully Developed Couette Flow Step 6: Verify solution by back-substituting into differential equations Given the solution (u,v,w)=(vy/h, 0, 0) Continuity is satisfied = 0 X-momentum is satisfied Fondamenti di Meccanica dei Continui 47

48 Example exact solution (Ex. 9-15) Fully Developed Couette Flow Finally, calculate shear force on bottom plate Shear force per unit area acting on the wall Note that τ w is equal and opposite to the shear stress acting on the fluid τ yx (Newton s third law). Fondamenti di Meccanica dei Continui 48

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