# Chapter 9: Differential Analysis

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1 9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems

2 Recall 9-1 Introduction (1) 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 9-1

3 9-1 Introduction (2) Example: incompressible Navier-Stokes equations We will learn: Physical meaning of each term How to derive How to solve 9-2

4 Step 9-1 Introduction (3) For example, how to solve? Analytical Fluid Dynamics (Chapter 9) Computational Fluid Dynamics (Chapter 15) 1 Setup Problem and geometry, identify all dimensions and parameters 2 List all assumptions, approximations, simplifications, boundary conditions 3 Simplify PDE s Build grid / discretize PDE s 4 Integrate equations Solve algebraic system of equations including I.C. s and 5 Apply I.C. s and B.C. s tosolve B.C s for constants of integration 6 Verify and plot results Verify and plot results 9-3

5 9-2 Conservation of Mass (1) 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 s) Theorem Differential CV and Taylor series expansions 9-4

6 9-2 Conservation of Mass (2) 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. 9-5

7 9-2 Conservation of Mass (3) Rewrite conservation of momentum Using divergence theorem, replace area integral with volume integral and collect terms Integral holds for ANY CV, therefore: 9-6

8 9-2 Conservation of Mass (4) 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 9-7

9 9-2 Conservation of Mass (5) 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 9-8

10 9-2 Conservation of Mass (6) 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 9-9

11 9-2 Conservation of Mass (7) After substitution, Dividing through by volume dxdydz Or, if we apply the definition of the divergence of a vector 9-10

12 9-2 Conservation of Mass (8) Alternative form Use product rule on divergence term 9-11

13 9-2 Conservation of Mass (9) There are many problems which are simpler to solve if the equations are written in cylindrical-polar i l l coordinates Easiest way to convert from Cartesian is to use vector form and definition of divergence operator in cylindrical coordinates 9-12

14 9-2 Conservation of Mass (10) 9-13

15 9-2 Conservation of Mass (11) 9-14

16 9-2 Conservation of Mass (12) Steady compressible flow Cartesian Cylindrical 9-15

17 9-2 Conservation of Mass (13) Incompressible flow and ρ = constant Cartesian Cylindrical 9-16

18 9-2 Conservation of Mass (14) 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 9-17

19 9-3 The Stream Function (1) Consider the continuity equation for an incompressible 2D flow Substituting the clever transformation ti Gives This is true for any smooth function ψ(x,y) 9-18

20 9-3 The Stream Function (2) Why do this? Single variable ψ replaces (u,v). Once ψ is known, (u,v) can be computed. Physical significance 1. Curves of constant t ψ are streamlines of the flow 2. Difference in ψ between streamlines is equal to volume flow rate between streamlines 9-19

21 9-3 The Stream Function (3) Physical Significance Recall from Chap. 4 that along a streamline Change in ψ along streamline is zero 9-20

22 Physical Significance 9-3 The Stream Function (4) Difference in ψ between streamlines is equal to volume flow rate between streamlines 9-21

23 9-4 Conservation of Linear Momentum (1) Recall CV form from Chap. 6 Body Force Surface Force σ ij = stress tensor Using the divergence theorem to convert area integrals 9-22

24 9-4 Conservation of Linear Momentum (2) 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) 9-23

25 9-4 Conservation of Linear Momentum (3) Alternate form of the Cauchy Equation can be derived by introducing (Chain Rule) Inserting these into Cauchy Equation and rearranging gives 9-24

26 9-4 Conservation of Linear Momentum (4) 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! 9-25

27 9-5 Navier-Stokes Equation (1) First step is to separate σ ij into pressure and viscous stresses σ xx σ xy σ xz pp 0 0 τ xx τ xy τ xz σ ij = σ yx σ yy σ yz = 0 p 0 + τ yx τ yy τ yz σ zx σ zy σ zz 0 0 p τ zx τ zy τ zz Situation not yet improved Viscous (Deviatoric) Stress Tensor 6 unknowns in σ ij 6 unknowns in τ ij + 1inP P, which means that we ve added 1! 9-26

28 9-5 Navier-Stokes Equation (2) (toothpaste) (paint) (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 Newtonian closure is analogous 9-27 to Hooke s Law for elastic solids

29 9-5 Navier-Stokes Equation (3) Substituting Newtonian closure into stress tensor gives Using the definition of ε ij (Chapter 4) 9-28

30 9-5 Navier-Stokes Equation (4) 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) 9-29

31 9-5 Navier-Stokes Equation (5) In addition to vector form, incompressible N-S equation can be written in several other forms Cartesian coordinates Cylindrical coordinates Tensor notation 9-30

32 9-5 Navier-Stokes Equation (6) Cartesian Coordinates Continuity X-momentum Y-momentum Z-momentum See page 431 for equations in cylindrical i l coordinates 9-31

33 9-5 Navier-Stokes Equation (7) Tensor and Vector notation offer a more compact form of the equations. Continuity Tensor notation 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 = VU V, 3 = W) 9-32

34 9-6 Differential Analysis Problems (1) 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) 9-33

35 9-6 Differential Analysis 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 Problems (2) 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 9-34

36 9-6 Differential Analysis Problems (3) 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 t of integration ti 6.Verify results 9-35

37 9-6 Differential Analysis Problems (4) 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 9-36

38 9-6 Differential Analysis Problems (5) No-slip boundary condition For a fluid in contact with a solid wall, the velocity of the fluid must equal that of the wall 9-37

39 9-6 Differential Analysis Problems (6) Interface boundary condition When two fluids meet at an interface, the velocity and shear stress must be the same on both sides If surface tension effects are negligible and the surface is nearly flat 9-38

40 9-6 Differential Analysis Problems (7) 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 9-39

41 9-6 Differential Analysis Problems (8) Example: Fully Developed Couette Flow (Ex. 9-15) 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 9-40

42 9-6 Differential Analysis Problems (9) Step 2: Assumptions and BC s Assumptions 1. Plates are infinite in x and z 2. Flow is steady, / t = 0 3. Parallel flow, V=0 4. Incompressible, Newtonian, laminar, constant t properties 5. No pressure gradient 6. 2D, W=0 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 9-41

43 9-6 Differential Analysis Problems (10) 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

44 9-6 Differential Analysis Problems (11) Step 3: Simplify, cont. Y-momentum 2, , Z-momentum 2,

45 9-6 Differential Analysis Step 4: Integrate X-momentum Problems (12) integrate integrate Z-momentum integrate 9-44

46 9-6 Differential Analysis Step 5: Apply ppy BC s Problems (13) y=0, u=0=c 1 (0) + C 2 C 2 = 0 y=h, u=v=c 1h 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 9-45

47 9-6 Differential Analysis Problems (14) 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 9-46

48 9-6 Differential Analysis Problems (15) Finally, calculate shear force on bottom plate Shear force per unit area acting on the wall Note that t τ w is equal and opposite to the shear stress acting on the fluid τ yx (Newton s third law). 9-47

49 9-6 Differential Analysis Problems (16) Computational Fluid Dynamics (CFD): Example

50 9-6 Differential Analysis Problems (17) Computational Fluid Dynamics (CFD): Example

51 9-6 Differential Analysis Problems (18) Computational Fluid Dynamics (CFD): Example

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