1. CFD Equations Chapter 1

2. Training Manual May 15, 2001 Inventory #001478 1-2 Navier-Stokes Equations, Conservation of Mass, and the Energy Equation Definition of the Equations The Continuity Equation The Stress-Strain Relation Forms of the Equations Important Properties Dimensionless Parameters Dimensionless Equations The Energy Equation Dimensionless Energy Equation Rotational Frames of Reference Swirl

3. Training Manual May 15, 2001 Inventory #001478 1-3 It is common to use vector and tensor notation to describe the equations compactly. Sometimes, within a single equation it is convenient to use different notation for different terms. Some ways to express the Continuity Equation (conservation of mass): Scalar Equation: where U,V,W are velocities in orthogonal x,y,z directions Vector form: where V is the vector of velocity Conservation of Mass (and notation…)

4. Training Manual May 15, 2001 Inventory #001478 1-4 Indicial notation - repeated subscript means summing the terms: u i represents velocities in the three x i directions The substantial derivative is particularly useful in describing transport. The operator is: and its use yields a fourth expression of continuity: Conservation of Mass… (con’t)

5. Training Manual May 15, 2001 Inventory #001478 1-5 Note that for the case of constant density: The means divergence of the velocity is zero in constant property flows. This is often checked as a condition of accuracy or convergence in computational fluids. It is fairly common for the continuity equation to serve as a link in determining the pressure in computational algorithms. Generally, you can assume that the Navier- Stokes equation provides the velocities in response to the pressure field. Remarks on Continuity

6. Training Manual May 15, 2001 Inventory #001478 1-6 The velocity or its gradient is put in terms of the pressure gradient in some fashion. In such a case, the density change must be put in terms of pressure. Introduce the Bulk modulus: So that This term is responsible for the rate at which sound waves will propagate. The higher the value of K, the faster the propagation of the wave. More Remarking

7. Training Manual May 15, 2001 Inventory #001478 1-7 Remarks (continued) In the literature, speed of sound is given by an expression quite similar to the Bulk modulus... The “bulk modulus parameter” used in FLOTRAN is specified by the user for incompressible transient flows: When the compressible formulation is used, the value is calculated by FLOTRAN based on the Ideal Gas Law

8. Training Manual May 15, 2001 Inventory #001478 1-8 The Navier-Stokes Equations come from applying the Conservation of Momentum to the flow of a Newtonian fluid (The characteristics of which will be described shortly!). Begin with the Momentum Equation - Newton’s Law of Motion Represents three equations for the three orthogonal directions I=1,2,3 (e.g.; x,y,z in Cartesian space) Acceleration (due to) Body Forces and Surface Forces Vector Expression Conservation of Momentum

9. Training Manual May 15, 2001 Inventory #001478 1-9 Types of Terms in the Navier-Stokes Equations Acceleration Terms: –Non-Linear –Continuity Equation Imbedded –Treated as advection transport Body Force Terms: –Gravity –Rotating Coordinate System –Effects of Magnetic Fields

10. Training Manual May 15, 2001 Inventory #001478 1-10 Types of Terms (continued) Surface Force Terms: –Normal Pressure (Mechanical, Thermodynamic) –Shear Stresses Treated as diffusion terms The “Navier-Stokes Equations” are the momentum equations as formulated for a Newtonian Fluid –Next, just what is a Newtonian Fluid?

11. Training Manual May 15, 2001 Inventory #001478 1-11 The three postulates of Stokes (1845) lead to the description of the Newtonian Fluid: –1. The fluid is continuous and isotropic. –2. The stress tensor is at most a linear function of the strain rate. –3. With zero strain, the deformation laws must reduce to the hydrostatic pressure condition. The resulting relationship is:  absolute viscosity second coefficient of viscosity (rarely considered) This relationship is valid for all gases and most common fluids. The Stress-Strain Relationship for the Newtonian Fluid

12. Training Manual May 15, 2001 Inventory #001478 1-12 The Second Coefficient of Viscosity Treatment Our Goal: Get rid of this irritating term... 1. Consider the fluid Incompressible. So, continuity reduces to the divergence of velocity, and the term containing vanishes. 2. Assume the term is small anyway and is neglected (this might not be true near shock waves). 3. Stokes Hypothesis. Based on requiring the thermodynamic and mechanical pressures to be the same...

13. Training Manual May 15, 2001 Inventory #001478 1-13 Mechanical Pressure is the average compression stress on an element of fluid. This in turn is a tensor invariant, so it can be expressed in the principle stress directions. From the expression for the principle stresses: Which leads to: So Stokes Hypothesis assumes away the problem... Second Coefficient (continued)

14. Training Manual May 15, 2001 Inventory #001478 1-14 Next, look at the acceleration terms: The last two terms are the velocity multiplied by the continuity equation, making them vanish. The resulting acceleration term allows a more compact statement of the equation Navier-Stokes Equations

15. Training Manual May 15, 2001 Inventory #001478 1-15 ALE Formulation The Momentum Equation (Navier-Stokes Equation) as presented assumes that the mesh is stationary. The Arbitrary Lagrangian Eulerian formulation (ALE) modifies the equations to account for mesh motion. This is required in Transient Fluid Structure Interaction problems when the fluid problem domain is changing with time.

16. Training Manual May 15, 2001 Inventory #001478 1-16 ALE Formulation - Momentum Equation The governing NS equations must be modified to reflect this mesh motion: Mesh velocity

17. Training Manual May 15, 2001 Inventory #001478 1-17 The Non-Dimensionalization of the Equations Put equations into comparative context. This aids in determining which terms are important, given some flow conditions and properties. –Properties of Fluid –Reference Conditions –Boundary Conditions –Dimensionless Parameters

18. Training Manual May 15, 2001 Inventory #001478 1-18 Express the conditions as non-dimensional multipliers of the reference conditions. The Basic Properties of the Fluid  Density  Absolute viscosity Second coefficient of viscosity  Coefficient of Thermal Expansion S Surface Tension Reference quantities: V o Magnitude of the reference velocity k Thermal conductivity C P Specific Heat at constant pressure C V Specific Heat at constant volume l Mean free path  o,  o, T o Reference values

19. Training Manual May 15, 2001 Inventory #001478 1-19 Relate the acceleration term to the viscous term. Neglect body forces such as gravity for the moment. This is appropriate for “high speed” gas flow, for example. Reynold’s Number: The D h is the hydraulic diameter for internal flow: Basic Properties (continued)

20. Training Manual May 15, 2001 Inventory #001478 1-20 Basic Properties (continued) Characteristic length for external flows –Chord length of airfoil –Distance from the leading edge The Reynolds number is the ratio of advection (transport by virtue of the velocity) to the transport by diffusion. Density and viscosity are relevant fluid properties D h and V are conditions of the problem.

21. Training Manual May 15, 2001 Inventory #001478 1-21 Kinematic Viscosity is the only property in the Kinematic Expression of the Reynold’s Number Some Kinematic Viscosity's (meter 2 /sec) - 20C Glycerin5.0E-4Kerosene2.5E-6 SAE30 Oil2.5E-4Water1.0E-6 SAE10 Oil1.0E-4Benzene7.0E-7 Air1.8E-5Mercury1.5E-7 Crude Oil1.0E-5 For a given set of conditions, the Kinematic viscosity varies amongst fluids as the inverse the Reynold’s Number does. Kinematic Viscosity

22. Training Manual May 15, 2001 Inventory #001478 1-22 Reference Conditions for Non-Dimensionalization Choose constant reference values for density and viscosity. Choose free stream velocity V o and a Length scale L. Relate the distance, velocity, pressure, and properties to reference values. The details of the derivations are provided in the Chapter 1 Appendix (the end of this chapter).

23. Training Manual May 15, 2001 Inventory #001478 1-23 Non-Dimensional Momentum Equation The Reynolds number Re signals the relative importance of the advection and the diffusion contributions. The Grashoff number Gr shows the relative importance of buoyancy effects.

24. Training Manual May 15, 2001 Inventory #001478 1-24 Non-Dimensional Energy Equation The Peclet number Pe is the product of the Reynolds number and the Prandtl Pr number and indicates the relative importance of the transport and conduction of energy Eckert Number

25. Training Manual May 15, 2001 Inventory #001478 1-25 Prandtl Number: Prandtl Numbers for various fluids: Mercury0.024Water7.0 Helium0.70Benzene7.4 Air0.72Ethyl Alcohol16 Liquid Ammonia2.0SAE30 Oil3500 Freon-123.7Glycerin12,000 Methyl Alcohol6.8 The Energy Equation (continued)

26. Training Manual May 15, 2001 Inventory #001478 1-26 Rotating Coordinates The governing equations of motion in a rotating reference frame with constant angular velocity. This is useful in the analysis of rotating machinery. –Let v be the velocity of an arbitrary point in a fluid with respect to the rotating coordinate frame which has a constant angular velocity . –Denote the position of the point, measured with respect to the origin of the rotating coordinate system, as r. Computational solutions for rotating coordinates entail solving for velocities relative to the rotating frame. Numerical difficulties can arise because terms involving the Coriolis and Centrifugal accelerations can be large. This leads to a modification of the pressure variable to include some of these terms.

27. Training Manual May 15, 2001 Inventory #001478 1-27 The vector form of the momentum equation in the rotating reference frame with constant viscosity is: The general equation in indicial notation: Equations in the Rotating Frame

28. Training Manual May 15, 2001 Inventory #001478 1-28 Formulation The rotating acceleration will take the form of additional source terms. Momentum equations in X,Y,Z Space Concentrating on the acceleration terms, denote the shear stress as shown for XYZ Directions:

29. Training Manual May 15, 2001 Inventory #001478 1-29 The magnitude of the source terms due to the rotation can present numerical difficulties. The centrifugal portion of the additional terms can be put into the previous definition of the pressure. Further Modification to the Pressure

30. Training Manual May 15, 2001 Inventory #001478 1-30 In the absence of rotation, this modified pressure is the same as defined earlier. With a stationary reference frame: With a rotating reference frame: Further Modification to the Pressure (continued)

31. Training Manual May 15, 2001 Inventory #001478 1-31 Further Modification to the Pressure (continued) The governing equations in terms of the velocity with respect to the rotating coordinate system and a modified pressure:

32. Training Manual May 15, 2001 Inventory #001478 1-32 A Rotating Test Case Flow in the annulus between two cylinders. –Inner cylinder rotates, the outer is stationary. Goal: calculate the static pressure at the inner wall. Stationary Frame Boundary Conditions –Angular velocity of 1 - counter-clockwise direction –Apply velocity magnitude of 1 on the inner circle. –Outer wall is stationary; apply pressure as zero

33. Training Manual May 15, 2001 Inventory #001478 1-33 A Rotating Test Case (continued) Rotating frame: –Velocity at the inner cylinder is stationary. –Outer wall moves with a velocity magnitude of 2 clockwise. –Modified pressure must be applied to the outer boundary.

34. Training Manual May 15, 2001 Inventory #001478 1-34 Equations of motion: The following simplifications are possible: The continuity equation yields: Exact Solution Flow Between Rotating Cylinders

35. Training Manual May 15, 2001 Inventory #001478 1-35 The radial velocity is zero at the inner and outer radii. Deduce that the gradient is also zero there and everywhere. Thus: The solution for velocity is: Solution (continued)

36. Training Manual May 15, 2001 Inventory #001478 1-36 Use this to get the solution for pressure. The pressure equation is thus: Integration yields: Evaluate C3 from pressure boundary condition. Solution (continued)

37. Training Manual May 15, 2001 Inventory #001478 1-37 Swirl Axisymmetric flow with a component normal to the axisymmetric plane is known as Swirl. Note that the flow between rotating cylinders can also be solved with the Swirl option! VZ Normal to this plane

38. Training Manual May 15, 2001 Inventory #001478 1-38 The Equations for Swirl Swirling flow exists when an axisymmetric flow pattern has an azimuthal flow component Conveniently described by cylindrical coordinates with zero gradients (velocity, pressure) in the azimuthal direction. In other words, add a swirl component to the axisymmetric equations. Useful in coal plant applications where flows have an added rotational component. Rotating machinery in axisymmetric geometry (e.g., spinning shaft). Swirl velocity boundary conditions: –Inlet component of swirl –Moving wall (rotating cylinder)

39. Training Manual May 15, 2001 Inventory #001478 1-39 Equations of Motion Cylindrical coordinates without  dependence. Swirl flow does effect the X-R solution. Swirl component loosely coupled to other components. Coordinate system directions r, ,z

40. Training Manual May 15, 2001 Inventory #001478 1-40 Momentum Equations

41. Training Manual May 15, 2001 Inventory #001478 1-41 Continuity Equation

42. Training Manual May 15, 2001 Inventory #001478 1-42 The Inner cylinder rotates rotates, outer stationary –z velocity and directional dependence vanishes –No time dependence, neglect gravity, constant density The analytical solution was previously attained... Swirl Example - Rotating Cylinder

43. Training Manual May 15, 2001 Inventory #001478 1-43 Chapter 1 - Appendix A What follows are some of the details of the nondimensionalization of the momentum and energy equations

44. Training Manual May 15, 2001 Inventory #001478 1-44 Term-by-Term Non-Dimensionalization Advection Terms: or Stress Terms: or

45. Training Manual May 15, 2001 Inventory #001478 1-45 The Pressure gradient term: Combine all of these terms, and move the constants to one place: or Term-by-Term Non-Dimensionalization (continued)

46. Training Manual May 15, 2001 Inventory #001478 1-46 Natural convection problems, with an absence of an identifiable free stream velocity, call for a reference velocity based on a Reynold’s Number of unity: Note that the second coefficient of viscosity has been neglected along with the body forces. Now that the form of the shear stress terms is known, for convenience use: So that: Term-by-Term Non-Dimensionalization (continued)

47. Training Manual May 15, 2001 Inventory #001478 1-47 Before turning to the Energy Equation, examine the Navier- Stokes equations for low speed flow cases where gravity is important (e.g., natural convection). It is observed that effect of density changes may be neglected in all terms except the body force terms. This is known as the Boussinesq approximation, and it is commonly used with the assumption of the following form for the density changes: where  is the thermal expansion coefficient Term-by-Term Non-Dimensionalization (continued)

48. Training Manual May 15, 2001 Inventory #001478 1-48 The Navier-Stokes equation becomes: At this point, a change of variables for pressure is invoked. The constant density head is included in the pressure term, which is now expressed in terms of a static pressure with reference pressure such as atmospheric pressure. Term-by-Term Non-Dimensionalization (continued)

49. Training Manual May 15, 2001 Inventory #001478 1-49 From now on, drop the “mod” in the designation of the static pressure. Note that since the density in other than the gravity term is taken as the reference density we see that for all the terms: Also shown is the expression for the gravitational acceleration. Term-by-Term Non-Dimensionalization (continued)

50. Training Manual May 15, 2001 Inventory #001478 1-50 Non-dimensionalize each term in the gravity component, noting that everything on the right hand side was multiplied by the inverse of the constants from the advection term... The Temperature... It is customary to non-dimensionalize the temperature in terms of a temperature differential from a reference temperature and a reference temperature delta: Term-by-Term Non-Dimensionalization (continued)

51. Training Manual May 15, 2001 Inventory #001478 1-51 Finally, put the gravity term into non-dimensional terms: And of course, the Grashoff number has been introduced: Navier-Stokes Equation for Boussenesq fluid: Term-by-Term Non-Dimensionalization (continued)

52. Training Manual May 15, 2001 Inventory #001478 1-52 Term-by-Term Non-Dimensionalization (continued) The equations simplify slightly for natural convection, because you can assume a Reynold’s Number of unity. For forced flow cases, it is natural to ask if the buoyancy terms are important. Usually, the Froude number is consulted:

53. Training Manual May 15, 2001 Inventory #001478 1-53 Without further ado... –where C p is the specific heat –and the last term is the viscous dissipation function It is interesting that the second coefficient of viscosity must be no lower than the value implied by Stokes hypothesis to ensure that the viscous dissipation term is positive. The Energy Equation

54. Training Manual May 15, 2001 Inventory #001478 1-54 The Energy Equation (continued) Non-dimensionally term by term: Pressure term:

55. Training Manual May 15, 2001 Inventory #001478 1-55 Diffusion or conduction term: Dissipation term: Next, collect the reference terms and find the right dimensionless parameters... The Energy Equation (continued)

56. Training Manual May 15, 2001 Inventory #001478 1-56 Following the strategy used in the Navier-Stokes Equation, multiply the entire equation by the inverse of the group of reference quantities on the left... Obviously, it is time to introduce more dimensionless relationships! The Energy Equation (continued)

57. Training Manual May 15, 2001 Inventory #001478 1-57 The Eckert Number looks at the relative strengths of kinetic energy and energy storage. Peclet Number The Peclet Number is the ratio of thermal transport by advection to thermal transport by diffusion. The Energy Equation (continued)

58. Training Manual May 15, 2001 Inventory #001478 1-58 Non-Dimensional Energy Equation Introduce these into the non-dimensionalized equation to get: All the terms are important for high speed gas flows. For flows below Mach=0.3, the Eckert number is small enough to validate neglecting the pressure term and the viscous dissipation term.

59. Training Manual May 15, 2001 Inventory #001478 1-59 Non-Dimensional Energy Equation (continued) To compare the relative strength of the conduction and dissipation terms, you introduce the Brinkman number. If it is significantly greater than unity, viscous dissipation should be considered: