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The error transport equation is derived from the governing equations by assuming that the discrete finite volume solution introduces a residual !

R that represents the amount by which the governing partial differential equation is not satisfied. Starting with the Navier-Stokes equations for a turbulent, real gas mixture in integral form, we subtract from it the approximate finite volume solution on a mesh of width h to obtain the ETE.

∂ species and the turbulence model employed, such as the k-epsilon (k-ε) model.

ε =! ! This error transport equation is expanded from that reported in the initial works by the author to include additional transport equations for the error in the partial density of each species comprising the mixture, error source terms due to the predicted error in turbulent mass diffusivity, which is computed from error in turbulent viscosity.

The inviscid portion of the residual required in Eqn. 8 is formed from the upwind contribution of the inviscid flux at each face of the control volume, which is derived using the Modified Equation approach [41, 50]. Following the work of Zhang et al. [40, 41], the flux at the face is computed from the flux evaluated at the states to the left and right of the face, plus the upwind contribution.

F!

( )

Q^{! =12}

(

^F!

( )

^{Q!L +F!}

( )

^Q!R

)

⁺¹²

^{( )}

^ΔF! (10)

Note that for a preconditioned system, the upwind flux would also include the preconditioning matrix. Starting with a simple explicit finite volume Roe scheme,

u_i^h,n+1

= u

_i^h,n

− Δt

Δx (

f_i+1/2^h,n

− f

_i−1/2^h,n

)

(11)

u_i^h,n+1

= u

_i^h,n

− Δt Δx

1

2

(

f_i+1^h,n

+ f

_i^h,n

) ⁻

¹²

^Δf

^i+1/2h,n

⁻

¹²

(

^fih,n

^{+ f}

^i−1h,n

) ⁺

¹²

^Δf

^i−1/2h,n

( )

(12)

we expand u_i^h,n+1, f_i+1^h,n, and f_i−1^h,n via Taylor series. Derivation of the modified equation is straightforward and becomes

∂u

∂t +∂f

∂x = 1

2Δx Δfi+1/2

h,n − Δfi−1/2

(

h,n

)

^−Δt₂ ^∂_∂t^{2u2 −Δx}₆^{2 ∂}_∂x^{3f3 −Δt}₆^{2 ∂}_∂t^{3u3 + O Δx}

⁽

^3,Δt3

⁾

(13) the leading terms of which are simply the upwind terms of the Roe flux at the cell faces.

R = 1

2Δx

(

Δf_i+1/2^{h ,n} − Δf_i−1/2^{h ,n}

)

(14)

Zhang et al. also noted more complex expressions for the residual may be formed by expanding the upwind terms in Eqn. 12 as well. Furthermore, in transient applications, where the ETE is solved concurrent with the Navier-Stokes equations, additional terms appear in the residual. For a finite volume scheme, the inviscid contribution to the residual at node i is formed from the upwind contribution from each of the edges incident to the node.

R^!_INV = −1

2 Δ!

F ⋅ ˆn

( )

^dA

∫∫

⁽¹⁵⁾

It should be noted that although Zhang’s approach was derived for a first order upwind scheme, the approach is directly applicable to a second-order MUSCL implementation of the Roe flux in which the states are reconstructed at each side of the face using gradient extrapolation from the neighboring control volumes. All applications presented were obtained using a second-order MUSCL scheme with Total Variation Diminishing (TVD) limiting.

The modified equation approach for the viscous terms results in high order derivatives that are difficult to form for practical calculations on unstructured meshes, which are emphasized in the applications that follow. As an alternative model, error in the viscous flux may be broken into two components: error in the velocity components u_i and temperature T, assuming µt is correct; and error in µ_t, assuming the velocity components u_i and temperature T are correct. Extending this model to real gas systems, we assume that the error in the viscous flux for the species transport equations may be represented in a similar manner: error in the species mass fractions y_i, assuming D_t is correct; and error in D_t , assuming the mass fractions y_i are correct. The difference in the viscous fluxes in Eqn. 8 may then be written as follows:

G! !

This leads to a diffusion term for the error and a production term or residual due to the predicted error in turbulent viscosity, thermal conductivity, and mass diffusivity.

! G !

( )

Q ^−G!

( )

^Q!h

( )

^{i ˆn =G !!}

( )

^{ε i ˆn −R!TURB (22)}

Equation 22 provides for a suitable diffusion mechanism for the error, as well as a production term, due to predicted errors in turbulent viscosity, conductivity, and diffusivity. These mechanisms were found to be essential in prior studies of boundary layer and shear layer flows [51], as inaccurate prediction of turbulent mixing is known to affect mean quantities. Thus the model is able to treat portions of the domain that are dominated by diffusion versus convection.

The error in turbulent viscosity is derived from the predicted errors in the turbulence model variables, from which the errors in turbulent conductivity and diffusivity follow given the turbulent Prandtl and Schmidt numbers. The error in turbulent viscosity for the k-ε model is given in Eqn. 23. In addition to k-ε, the ETE solver in CRISP CFD supports solutions obtained using the Spalart-Allmaras and k-ω turbulence models [53].

2 1

The final viscous ETE is therefore

∂ variables, from which the errors in turbulent conductivity and diffusivity follow given the turbulent Prandtl and Schmidt numbers. The turbulent production term in Eqn. 24 provides an error generation mechanism that accounts for error in momentum and energy transfer due to over- or under-predicted mixing. An error tensor is computed in a manner analogous to the stress tensor, in which the error in eddy viscosity appears in place of the effective viscosity.

ε

, the error in turbulent thermal conductivity may be obtained assuming a constant turbulent Prandtl number. For details on the turbulent residual the reader is referred to Ref. 51.

The formation of

µt

ε

itself is dependent on the turbulence model employed. To date we have implemented the one-equation Spalart-Allmaras model and the two-equation k-ε and k-ω models in the ETE solver. For each, the transport equations for the relevant turbulence quantities are

ε

are derived using procedures described in the next section.

In document Sector FARMAcéutico en Uruguay (página 24-31)