3. FORMS OF GOVERNING EQUATIONS IN CFD

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Nigel Rich
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1 3. FORMS OF GOVERNING EQUATIONS IN CFD 3.1. Governing and model equations in CFD Fluid flows are governed by the Navier-Stokes equations (N-S), which simpler, inviscid, form is the Euler equations. For studying CFD, we often use simple model equations. These share the same properties as the real governing equations, but are simpler to program and to analyze. In this section, various forms of the N-S and Euler equations will be reviewed, along with the most typical model equations as well Conservative vs. non-conservative forms of the Navier-Stokes equations The Navier-Stokes (N-S) equations are the fundamental governing equations of fluid flows, valid for any type of flow: let it be high-speed or low-speed, viscous or inviscid, compressible or incompressible, steady or unsteady, etc. They are based on 3 fundamental physical principles: 1) Conservation of mass (Continuity law) 2) Conservation of momentum (Newton s 2 nd law, F=m.a) 3) Conservation of energy [n1] Note: Using these fundamental principles, one can derive the Navier-Stokes equations in 4 different ways: Flow model Type Leads to: eq. type eq. form finite control volume fixed integral conservation (1) moving integral non-conservation (2) infinitesimally small volume fixed differential conservation (3) moving differential non-conservation (4) Note that it is extremely important for simulating compressible fluid flows that - what type (integral or differential) and - what form (conservation or non-conservation) are the governing equations. The reasons will become clear later. 1

2 Also note that the 4 forms are mathematically equivalent. Historically, all 4 types have been around for about 150 years before the classification into conservation and non-conservation forms become important. This classification was introduced around 1980, exactly for the purposes of CFD. Application of the three fundamental principles (of conservation of mass, momentum and energy) to any of the four flow models above leads to a set of continuity, momentum and energy equations. The detailed derivation for some of these models have been shown in Chapter 2, and in this Chapter, we are going to show the final equations for flow models (3) and (4) from the above table, i.e. the differential conservation and non-conservation forms. Continuity equations: [n2] Conservation form (case #3): Non-conservation form (case #4): Momentum equations: [n3] Conservation form (case #3): x: y: z: Non-conservation form (case #4): x: y: z: 2

3 Energy equations: [n4] Conservation form (case #3): Non-conservation form (case #4): In the above equations: [n5] [n6] body force per unit mass acting on fluid element, i.e. weight (due to gravity), electric or magnetic force. - rate of volumetric heat addition per unit mass, i.e. would be the volumetric heating of fluid element via absorption or emission of radiation - internal energy due to random molecular motion, per unit mass 3

4 - stress in the j direction exerted on a plane perpendicular to the i axis. [n7] For Newtonian fluids which means that the shear stress in the fluid is proportional to the time rate of strain (velocity gradient) the following definitions of stresses work: [n8] with: - molecular viscosity coefficient, which is function of T and p. The dependency on p is negligible, except at very low or very high pressures. The dependency on T is expressed via Sutherland s law: [n9] Gas C1 x 10 6 [kg/(m.s.k 0.5 )] C2 [K] Air CO CO H N O

5 - second viscosity coefficient, which was assumed by Stokes to equal to (This is yet to be proved) [n10] - thermal conductivity, which is defined synonymously as viscosity, i.e. it expresses proportionality between temperature gradient and heat conduction. It depends also on p and T, although the dependency on p is negligible. For air below 2,000 K: [n11] or via the Prandtl number: [n12] Note that the Prandtl number is defined as [n13]: Diffusion of momentum by viscosity Pr = = Diffusion of heat by conduction Also: - local space derivative, a vector operator of the form: [n14] - local time derivative, i.e. time rate of change of a variable at a fixed point in space. - time rate of change of a variable of the given fluid element as it moves through space. Definition for the specific example of density would be: 5

6 [n15] Closing equations: Note that we have 5 equations (1 continuity, 3 momentum, 1 energy) but 7 unknowns (, u, v, w, p, e, T), i.e. we need 2 more equations to close the system of equations. These are: Thermal equation of state (assuming perfect gas, i.e. negligible intermolecular forces): [n16] Caloric equation of state (assuming calorically perfect gas, i.e. with constant specific heats): [n17] Note: Viscous flow is defined as one where ANY of the following transport phenomena are included [n18]: 6

7 - friction (viscosity) - thermal conduction included in the above equations - mass diffusion not discussed in this course means concentration gradients of different chemical species in the flow, e.g, non-homogenous mixture of non-reacting gases, chemically reacting gases (like in hypersonic flows), different reaction rates at different locations, etc Flux vector formulation of the N-S equations The most convenient form of the above equations for usage in CFD (i.e. for programming convenience) is the flux vector formulation. What is a flux? We can observe that in the conservation forms of the N-S equations, the following flux terms appear [n19]: mass flux momentum flux, x-component momentum flux, y-component momentum flux, z-component flux of internal energy flux of total energy It can be observed that it is possible to create vectors of these fluxes and to rewrite the N-S equations in the following - and from programming point of view very convenient - form: 7

8 [n20] 8

9 Conservative vs. primitive variables The solution vector in the above vector formulation contains the so-called conservative or flux variables, i.e. (, (u), (v), (w) and (et). However, for analyzing CFD results, one is interested in visualizing the so-called primitive variables, i.e. (, (u), (v), (w) and (et). Once the solution vector of the conservative variables is known, the primitive variables can be easily obtained as: u v w u v w 2 V e u v w e 2 Note that the N-S equations can be expressed in vector formulation directly in terms of the primitive variables too. This would lead to the non-conservative form of the differential equations, which, as will be seen in the next section, is not the preferred form for compressible flows Advantages of the conservative form of N-S equations Beside the programming convenience, the flux vector formulation of the N-S equations has another major advantage, which is extremely important for compressible flows. This is associated with the meaning of the word conservative. The word conservative means to preserve (or to conserve ) something. In CFD, the variables written in conservative form preserve such important property, which they lose when written in their primitive form. This is their ability to preserve their differentiability in such flows, in which they cannot be differentiated any more if they were written in their primitive form. This can occur in such flows phenomena, in which a discontinuity (i.e. a sharp jump in the flow variables, or mathematically, the loss of continuisness of a function) occurs, such as for example a shock wave. Since the N-S equations are differential equations, the terms in them should ideally be differentiable, i.e. based on continuous functions. For this reason, the ability to preserve or to conserve the continuissness of a variable in its conservative form is a great 9

10 advantage for CFD programmers, who do build upon this property when designing CFD codes. Why can the N-S equations written in terms of the conservative (or flux) variables handle discontinuities better than in terms of primitive variables? Consider a shock wave and the variation of primitive or conservative variables through it: [n21] Expression relating states (1) and (2): In other words, while there are discrete (i.e. sharp) jumps in the flow variables when we monitor them in term of primitive variables, these disappear when we monitor them in terms of conservative variables.. As a result of this handling characteristic, schemes based on the conservative forms resolve shock waves more clearly, i.e. with less wiggles (oscillations in terms of the primitive variables) around shock waves or sharp discontinuities. They are also more stable (robust) than the non-conservative formulations, which often fail to converge or simply blow up when detecting shock waves. 10

11 An added very important consequence of employing a conservative form is that it captures the location and speed of the shock wave (if it is moving) correctly. This is a clear benefit over non-conservation forms, which consistently under-predict shock location and shock speed. Thus, the 5 major advantages of employing the conservative form of the governing equations for compressible flows are: 1) Programming convenience 2) Handling of discontinuities 3) More stable solution of flows with shock waves 4) Less wiggles around shock waves Good shock 5) Correct shock location handling 6) Correct shock speed qualities Note: the above arguments are true for any sharp discontinuities in the flow variables, which are governed by the conservation laws. Shock waves are just one prominent example of sharp discontinuities, which are associated with compressible supersonic flows. In other words, any fluid flow with sharp discontinuities is better resolved when the flow properties are solved in terms of the conserved variables, rather than primitive variables Euler equations Inviscid flow is defined as one where dissipative transport phenomena of: - friction (viscosity) - thermal conductivity are all neglected - mass diffusion This simply means dropping the terms involving friction and thermal conduction from the N-S equations, i.e. in flux vector form [n22]: Euler equations 11

12 Note: German scientist Leonhard Euler derived the continuity and momentum equations in 1753 for an inviscid fluid, and although he did not deal with the energy equation (since Thermodynamics arrived nearly a century later), we include the energy equation nowadays in what we call Euler equations Model equations The N-S and Euler equations are very complex partial differential equations, which up to date could not be solved analytically and require considerable effort to solve them numerically too. Because of their complexity, it is common in studying CFD to employ simple model equations with similar behaviour, such as [n23]: Laplace s equation (2D) : (1) Poisson s equation (2D): (2) Steady heat diffusion (1D): (3) Unsteady heat conduction (2D): (4) Stokes s problem (1D): (5) Linear advection (1D): (6) Wave equation (1D): (7) - Burger s equation (1D): (8) Scalar conservation law (1D): (9) 12

13 In particular, we are going to use equations (3), (4) and (9) for studying CFD methods for compressible flows. Why? Steady heat diffusion is excellent model to study the discretization of the diffusion, i.e. the viscous terms in the N-S equation. Unsteady heat conduction is a combination of diffusion and conduction, and hence nicely models the mixture of two types of terms. These two are excellent model equations to illustrate the implementation of the Finite Volume Method, which is the most popular method in most commercial CFD codes such as Vectis, ANSYS-CFX, FLUENT or OpenFOAM. The scalar conservation law concerns conduction only and as such, is an excellent model for the Euler equations. Scalar conservation law can model simple compression waves, shock waves and contact discontinuities, i.e. phenomena typical of compressible flows. 13

Differential relations for fluid flow

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