2. Governing Equations and Closure Structure

2.1. Introduction

Stream is a parallel Computational Fluid Dynamics (CFD) code with a wide variety of applications and can be used on any hardware. Its versatility lies in an all-speed methodology to handle all-speed flows which allows a seamless computation of flows involving a wide range of Mach numbers, i.e., incompressible to hypersonic in a unified manner. It can be used for a wide variety of applications involving multiphase, compressible/incompressible, laminar/turbulent flows, with or without chemical reactions and phase change. For handling complex geometries, it allows generalized unstructured grids with arbitrary polyhedral cells. One of the unique features of Stream is a powerful programming framework called Loci which is designed to reduce the complexity of assembling large-scale finite-volume applications as well as the integration of multiple applications in a multidisciplinary environment.

Various aspects of the Stream solver are available in references [KTWS06, TW12, TWS04, TWS05, WT04]. This chapter establishes the conservation laws, constitutive relations, thermodynamic constraints, and closure structure used throughout the guide. The remainder of the guide explains how those equations are solved by the segregated pressure-based method and how the required turbulence, reacting-flow, and other physical closures are constructed.

2.2. Notation Conventions

Notation

Scalars (e.g. pressure, density, temperature, species mass fractions) are italic letters: \(p, \rho, T, Y_k\).

Vectors (e.g. velocity, heat‐flux, diffusion velocity) carry an arrow: \(\vec{u}=(u_1,\,u_2,\,u_3), \vec{q}, \vec{V}_k\).

Second‐order tensors (stress, rate‐of‐strain, SGS stress) use a double‐underline: \( \Matrix{\tau},\quad \Matrix{S},\quad \Matrix{\tau}_{SGS}\).

Unit vectors use hats, e.g. \(\hat{\vec{n}}\).

Differential operators are upright: \(\nabla, \nabla\cdot, \partial\).

Tensor products use \(\otimes\): \(\vec{u}\otimes\vec{u}\).

The same notation policy is used consistently across all chapters.

2.3. Instantaneous Governing Equations

Below is a summary of the common-core equations governing fluid flow and heat transfer in turbulent, compressible, reacting flows. Generic conservative source placeholders are included so the equations can also represent coupled models such as Lagrangian particles, porous media, phase change, radiation, or user-specified volumetric forcing. Setting these source terms to zero gives the uncoupled carrier-flow equations. The derivation of these equations, details regarding the constitutive relations used and the various turbulence modeling assumptions employed can be found in several sources (e.g., White[Whi91], Wilcox[Wil00], Poinsot and Veynante[PV05], etc.). Using compact vector/tensor notation for this first summary, the instantaneous form of continuity, momentum, energy, and species transport equations can be written as follows:

Continuity equation:

(2.3.1)\[\frac{\partial \rho}{\partial t} + \nabla \cdot \left( \rho \vec{u} \right) = S_{\rho}\]

Momentum equation:

(2.3.2)\[\frac{\partial (\rho \vec{u})}{\partial t} + \nabla \cdot (\rho \vec{u} \otimes \vec{u}) = -\nabla p + \nabla \cdot \Matrix{\tau} + \vec{S}_{\rho\vec{u}}\]

Energy equation:

Total enthalpy form:

(2.3.3)\[\frac{\partial}{\partial t} \left( \rho H - p \right) + \nabla \cdot \left( \rho \vec{u} H \right) = -\nabla \cdot \vec{q} + \nabla \cdot \left( \Matrix{\tau} \cdot \vec{u} \right) + S_{\rho H}\]

Total energy form:

(2.3.4)\[\frac{\partial}{\partial t} \left( \rho E \right) + \nabla \cdot \left( \rho \vec{u} H \right) = -\nabla \cdot \vec{q} + \nabla \cdot \left( \Matrix{\tau} \cdot \vec{u} \right) + S_{\rho E}\]

Temperature form:

(2.3.5)\[\begin{split}\begin{aligned} \rho c_p \frac{D T}{D t} - \beta_T T \frac{D p}{D t} &= \nabla\cdot(\lambda\nabla T) -\sum_{k=1}^{N_s}\vec{J}_k\cdot\nabla h_k+\Phi \\ &\quad+S_{\rho e} -\sum_{k=1}^{N_s}h_k(\dot{\omega}_k+S_{\rho Y_k}). \end{aligned}\end{split}\]

Species transport equation:

(2.3.6)\[\frac{\partial}{\partial t} \left( \rho Y_k \right) + \nabla \cdot \left( \rho \vec{u} Y_k \right) = -\nabla \cdot \left( \rho Y_k \vec{V}_k \right) + \dot{\omega}_k + S_{\rho Y_k}, \quad k = 1,\ldots,N_s\]

Species Conservation Constraints

The species variables and fluxes are constrained by mixture conservation:

(2.3.7)\[\sum_{k=1}^{N_s}Y_k=1, \qquad \sum_{k=1}^{N_s}\rho Y_k\vec{V}_k=\vec{0}, \qquad \sum_{k=1}^{N_s}\dot{\omega}_k=0, \qquad \sum_{k=1}^{N_s}S_{\rho Y_k}=S_\rho .\]

All \(N_s\) species equations are solved. Their sum gives continuity when the species sources and fluxes satisfy these constraints. Conservative face reconstruction and diffusion enforce the corresponding discrete flux identities in Eqs. (13.3.1.7) and (13.3.2.3).

Note

The temperature equation uses the additive ideal-mixture relation \(h=\sum_kY_kh_k(T,p)\). The species enthalpies can describe real fluids and retain pressure dependence. For an ideal-gas mixture, \(\beta_T T=1\) and \(\nabla h_k=c_{p,k}\nabla T\), as discussed in Species Enthalpy Diffusion. The temperature derivation retains all species balances and the continuity source \(S_\rho\). The distinction between the continuum temperature equation and a finite energy balance is discussed in Temperature And Discrete Energy Conservation. Equivalent continuum forms need not give the same discrete energy balance.

For a more detailed equation-by-equation summary, including terms such as species-enthalpy diffusion, buoyancy, Lagrangian-particle mass/momentum/energy exchange, non-inertial-frame sources, and the distinction between continuum and solver-rewritten forms, see the Appendix on detailed governing equations. The frame kinematics and the resulting apparent-force terms are derived separately in the non-inertial reference-frame appendix.

In the above equations, \(t\) is time, \(\rho\) is density, \(\vec{u}\) is the velocity vector, \(p\) is pressure, \(Y_k\) is the mass fraction of species \(k\) (out of \(N_s\) species), \(\Matrix{\tau}\) is the viscous stress tensor, \(\vec{q}\) is the total heat-flux vector, and \(\vec{V}_k\) is the diffusion velocity of species \(k\). The total heat flux is defined in Eq. (2.3.1.5); in multispecies flows it includes both Fourier conduction and enthalpy carried by species diffusion. The source terms \(S_{\rho}\), \(\vec{S}_{\rho\vec{u}}\), \(S_{\rho H}\), \(S_{\rho E}\), and \(S_{\rho Y_k}\) are conservative volumetric sources. For example, evaporating or boiling particles give \(S_{\rho}=\sum_k S_{\rho Y_k}\), drag contributes to \(\vec{S}_{\rho\vec{u}}\), and particle/fluid heat exchange contributes to the energy source.

In Eq. (2.3.5), \(D/Dt=\partial/\partial t+\vec{u}\cdot\nabla\) is the material derivative, \(c_p=(\partial h/\partial T)_{p,Y}\) is the mixture specific heat at fixed pressure and composition, \(\beta_T=-\rho^{-1}(\partial\rho/\partial T)_{p,Y}\) is the isobaric thermal expansion coefficient, \(\Phi=\Matrix{\tau}:\nabla\vec{u}\) is the viscous-dissipation function. The species enthalpies \(h_k(T,p)\) are defined in Eq. (16.1.4), with units of J/kg of species \(k\). The conservative static-energy source follows from subtracting the carrier kinetic-energy balance:

(2.3.8)\[S_{\rho e} =S_{\rho E}-\vec{u}\cdot\vec{S}_{\rho\vec{u}} +\frac12|\vec{u}|^2S_\rho.\]

The conversion of model sources is detailed in Species Balances And The Temperature Source. The symbol \(\beta_T\) is distinct from the pressure under-relaxation factor \(\alpha_p\) used in the numerical-method chapters. The total enthalpy \(H\) is given by

(2.3.9)\[H = h + \frac{1}{2} \vec{u}\cdot\vec{u}\]

where \(h=e+p/\rho\) is the specific enthalpy, \(e\) is the specific internal energy, and the caloric closure may be written \(h=h(p,T,\boldsymbol{Y})\).

The total enthalpy is related to the total energy by:

(2.3.10)\[H = E + p/\rho\]

For an ideal-gas mixture, \(h\) is related to the temperature (\(T\)) by

(2.3.11)\[h = \sum_{k=1}^{N_s} Y_k \left(h_{0,k} + \int_{T_0}^{T} c_{p,k}\, dT \right)\]

where \(h_{0,k}\) is the enthalpy at the reference temperature \(T_0\) and \(c_{p,k}\) is the specific heat (for constant pressure processes) for species k. An equation of state is required to relate the density to the thermodynamic variables \(\rho = \rho(p,T,Y_k)\).

For an ideal-gas mixture, we use the following:

(2.3.12)\[p = \rho R_m T\]

where \(R_m=\sum_{k=1}^{N_s} Y_k R_k\) is the mixture gas constant.

2.3.1. Constitutive Relations for Newtonian Fluids

For the momentum equations, the constitutive relation between stress and strain rate for a Newtonian fluid is used to relate the components of the stress tensor to velocity gradients:

(2.3.1.1)\[\tau_{ij} = \mu \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) - \frac{2}{3} \mu \frac{\partial u_k}{\partial x_k} \delta_{ij} = 2 \mu \left( S_{ij} - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij} \right)\]

where \(\mu\) is the dynamic viscosity, \(S_{ij}\) is the rate-of-strain tensor, and \(\delta_{ij}\) is the Kronecker delta. This form invokes Stokes’ hypothesis for the isotropic dilatational stress. A more general Newtonian constitutive law may retain an independent bulk-viscosity coefficient.

For the species equations, a raw molecular diffusive mass flux for species k can be approximated using Fick’s law:

(2.3.1.2)\[\widehat{\vec{J}}_k = -\rho D_k \nabla Y_k\]

Here, \(D_k\) is the mass-diffusion coefficient of species k in the mixture. Independent raw Fick fluxes do not generally satisfy the mass-averaged constraint \(\sum_k\vec{J}_k=\mathbf{0}\) when the diffusivities differ. A mixture-averaged closure commonly applies the block correction

(2.3.1.3)\[\vec{J}_k \equiv \rho Y_k\vec{V}_k = \widehat{\vec{J}}_k - Y_k\sum_{\ell=1}^{N_s}\widehat{\vec{J}}_\ell, \qquad \sum_{k=1}^{N_s}\vec{J}_k=\mathbf{0}.\]

The accepted corrected \(\vec{J}_k\), rather than the independent raw flux, is the conservative species diffusion flux. When a raw diffusivity is specified through a Schmidt number,

(2.3.1.4)\[Sc_k = \frac{\mu}{\rho D_k}, \qquad D_k = \frac{\mu}{\rho Sc_k}, \qquad \widehat{\vec{J}}_k = -\frac{\mu}{Sc_k}\nabla Y_k .\]

Thus, \(D_k\) has units of area per time, while the coefficient \(\rho D_k=\mu/Sc_k\) multiplying \(\nabla Y_k\) has the units required for a diffusive mass flux.

For the energy equation, the total heat flux vector consists of a heat diffusion term obtained from Fourier’s law and a term associated with the diffusion of species with different enthalpies (which applies to a multi-species fluid):

(2.3.1.5)\[\vec{q} = -\lambda \nabla T + \sum_{k=1}^{N_s} h_k \vec{J}_k = -\lambda \nabla T + \rho \sum_{k=1}^{N_s} h_k Y_k \vec{V}_k .\]

Here, \(\lambda\) is the thermal conductivity and \(h_k\) is the specific enthalpy of species k. Substituting the raw Fick approximation and the zero-sum correction gives

(2.3.1.6)\[\vec{q} = -\lambda \nabla T - \rho \sum_{k=1}^{N_s} h_k D_k \nabla Y_k + \rho \left(\sum_{k=1}^{N_s} h_kY_k\right) \left(\sum_{\ell=1}^{N_s} D_\ell\nabla Y_\ell\right)\]

or, with the Schmidt-number specification,

(2.3.1.7)\[\vec{q} = -\lambda \nabla T - \sum_{k=1}^{N_s} h_k\frac{\mu}{Sc_k}\nabla Y_k + \left(\sum_{k=1}^{N_s} h_kY_k\right) \left(\sum_{\ell=1}^{N_s}\frac{\mu}{Sc_\ell}\nabla Y_\ell\right).\]

If all species use the same diffusivity, the raw fluxes already sum to zero because \(\sum_k\nabla Y_k=\mathbf{0}\), so the correction vanishes.

The laminar Prandtl number is defined as

(2.3.1.8)\[Pr_L = \frac{c_p \mu}{\lambda}\]

where \(c_p\) is the specific heat at constant pressure.

2.4. Closure Structure and Later Models

The conservation laws above define the common carrier-flow problem, but they are not closed until constitutive, thermodynamic, and model relations supply the stresses, heat and species fluxes, material properties, and source terms. The closure hierarchy used throughout this guide is:

Unclosed quantity or process	Closure supplied
Density, temperature, internal energy, enthalpy, and sound speed	An equation of state and caloric model
Viscous stress	Newtonian constitutive relation
Heat flux	Fourier conduction plus enthalpy carried by species diffusion
Species diffusion	A diffusion model satisfying the mixture flux constraint
Reynolds stresses and turbulent scalar fluxes	The RANS closures in Chapter 11
Filtered subgrid stresses	The LES and hybrid closures in Chapter 12
Chemical production and heat release	The finite-rate chemistry formulation in Chapter 13 or the flamelet formulation in Chapter 14
Moving-grid geometric terms	The geometric-conservation-law appendix
Non-inertial apparent forces	The non-inertial reference-frame appendix

The detailed Reynolds/Favre averaging that introduces the RANS closure terms is developed with the RANS models, rather than being interposed between the instantaneous equations and the numerical method. Likewise, spatial filtering and the subgrid-stress derivation are developed with the LES and hybrid models. This keeps the present chapter as the reference for the underlying conservation laws while preserving the complete averaging and filtering derivations where they are used.

Governing-Equation Reference

The pressure-based method does not replace the conservation laws with a different physical model. It solves their discretized forms while using pressure correction to couple momentum and continuity. Thermodynamic, turbulence, chemistry, multiphase, and other models enter by closing properties, fluxes, and source terms. The numerical-method chapters may rewrite these balances algebraically, but each rewriting must remain consistent with the equations and constraints collected here.

A short conceptual introduction to the declarative framework used to assemble the solver is provided in the Loci framework appendix.

2.5. References

[KTWS06]

R. Kamakoti, S. Thakur, J. Wright, and W. Shyy. Validation of a new parallel all-speed cfd code in a rule-based framework for multidisciplinary applications. In 36th AIAA Fluid Dynamics Conference and Exhibit. June 2006.

[PV05]

T. Poinsot and D. Veynante. Theoretical and Numerical Combustion. R.T. Edwards, 2nd edition, 2005.

[TW12]

S. Thakur and J. Wright. An all-speed solver for unsteady reacting and non-reacting flows using a rule-based framework. In 42nd AIAA Fluid Dynamics Conference. 2012.

[TWS04]

S. Thakur, J. Wright, and W. Shyy. A pressure-based algorithm for reacting flows with finite rate chemistry. In 34th AIAA Fluid Dynamics Conference. June 2004.

[TWS05]

S. Thakur, J. Wright, and W. Shyy. An algorithm for chemically reacting flows on generalized grids using a rule-based framework. In 43rd AIAA Conference. 2005.

[Whi91]

F.M. White. Viscous Fluid Flow. McGraw-Hill, 2nd edition, 1991.

[Wil00]

D.C. Wilcox. Turbulence Modeling for CFD. DCW Industries, 2nd edition, 2000.

[WT04]

J. Wright and S. Thakur. Development of a pressure-based combustion code in the loci framework. In NASA MSFC Spring Fluids Workshop. April 2004.