
By default, FLUENT stores discrete values of the scalar at the cell centers ( and in Figure 25.2.1). However, face values are required for the convection terms in Equation 25.22 and must be interpolated from the cell center values. This is accomplished using an upwind scheme.
Upwinding means that the face value is derived from quantities in the cell upstream, or "upwind,'' relative to the direction of the normal velocity in Equation 25.22. FLUENT allows you to choose from several upwind schemes: firstorder upwind, secondorder upwind, power law, and QUICK. These schemes are described in Sections 25.3.1 25.3.1.
The diffusion terms in Equation 25.22 are centraldifferenced and are always secondorder accurate.
FirstOrder Upwind Scheme
When firstorder accuracy is desired, quantities at cell faces are determined by assuming that the cellcenter values of any field variable represent a cellaverage value and hold throughout the entire cell; the face quantities are identical to the cell quantities. Thus when firstorder upwinding is selected, the face value is set equal to the cellcenter value of in the upstream cell.

Firstorder upwind is available in the pressurebased and densitybased solvers.

PowerLaw Scheme
The powerlaw discretization scheme interpolates the face value of a variable, , using the exact solution to a onedimensional convectiondiffusion equation
where and are constant across the interval . Equation 25.31 can be integrated to yield the following solution describing how varies with :
where
=  
= 
The variation of between and is depicted in Figure 25.3.1 for a range of values of the Peclet number. Figure 25.3.1 shows that for large Pe, the value of at is approximately equal to the upstream value. This implies that when the flow is dominated by convection, interpolation can be accomplished by simply letting the face value of a variable be set equal to its "upwind'' or upstream value. This is the standard firstorder scheme for FLUENT.
If the powerlaw scheme is selected, FLUENT uses Equation 25.32 in an equivalent "power law'' format [ 277], as its interpolation scheme.
As discussed in Section 25.3.1, Figure 25.3.1 shows that for large Pe, the value of at is approximately equal to the upstream value. When Pe=0 (no flow, or pure diffusion), Figure 25.3.1 shows that may be interpolated using a simple linear average between the values at and . When the Peclet number has an intermediate value, the interpolated value for at must be derived by applying the "power law'' equivalent of Equation 25.32.

The powerlaw scheme is available in the pressurebased solver and when solving additional scalar equations in the densitybased solver.

SecondOrder Upwind Scheme
When secondorder accuracy is desired, quantities at cell faces are computed using a multidimensional linear reconstruction approach [ 21]. In this approach, higherorder accuracy is achieved at cell faces through a Taylor series expansion of the cellcentered solution about the cell centroid. Thus when secondorder upwinding is selected, the face value is computed using the following expression:
(25.34) 
where and are the cellcentered value and its gradient in the upstream cell, and is the displacement vector from the upstream cell centroid to the face centroid. This formulation requires the determination of the gradient in each cell, as discussed in Section 25.3.3. Finally, the gradient is limited so that no new maxima or minima are introduced.

Secondorder upwind is available in the pressurebased and densitybased solvers.

CentralDifferencing Scheme
A secondorderaccurate centraldifferencing discretization scheme is available for the momentum equations when you are using the LES turbulence model. This scheme provides improved accuracy for LES calculations.
The centraldifferencing scheme calculates the face value for a variable ( ) as follows:
where the indices 0 and 1 refer to the cells that share face , and are the reconstructed gradients at cells 0 and 1, respectively, and is the vector directed from the cell centroid toward the face centroid.
It is well known that centraldifferencing schemes can produce unbounded solutions and nonphysical wiggles, which can lead to stability problems for the numerical procedure. These stability problems can often be avoided if a deferred approach is used for the centraldifferencing scheme. In this approach, the face value is calculated as follows:
(25.36) 
where UP stands for upwind. As indicated, the upwind part is treated implicitly while the difference between the centraldifference and upwind values is treated explicitly. Provided that the numerical solution converges, this approach leads to pure secondorder differencing.

The central differencing scheme is available only in the pressurebased solver.

Bounded Central Differencing Scheme
The central differencing scheme described in Section 25.3.1 is an ideal choice for LES in view of its meritoriously low numerical diffusion. However, it often leads to unphysical oscillations in the solution fields. In LES, the situation is exacerbated by usually very low subgridscale turbulent diffusivity. The bounded central differencing scheme is essentially based on the normalized variable diagram (NVD) approach [ 201] together with convection boundedness criterion (CBC). The bounded central differencing scheme is a composite NVDscheme that consists of a pure central differencing, a blended scheme of the central differencing and the secondorder upwind scheme, and the firstorder upwind scheme. It should be noted that the firstorder scheme is used only when the CBC is violated.

The bounded central differencing scheme is the default convection scheme for LES. When you select LES, the convection discretization schemes for all transport equations are automatically switched to the bounded central differencing scheme.


The bounded central differencing scheme is available only in the pressurebased solver.

QUICK Scheme
For quadrilateral and hexahedral meshes, where unique upstream and downstream faces and cells can be identified, FLUENT also provides the QUICK scheme for computing a higherorder value of the convected variable at a face. QUICKtype schemes [ 202] are based on a weighted average of secondorderupwind and central interpolations of the variable. For the face in Figure 25.3.2, if the flow is from left to right, such a value can be written as
(25.37) 
in the above equation results in a central secondorder interpolation while yields a secondorder upwind value. The traditional QUICK scheme is obtained by setting . The implementation in FLUENT uses a variable, solutiondependent value of , chosen so as to avoid introducing new solution extrema.
The QUICK scheme will typically be more accurate on structured grids aligned with the flow direction. Note that FLUENT allows the use of the QUICK scheme for unstructured or hybrid grids as well; in such cases the usual secondorder upwind discretization scheme (described in Section 25.3.1) will be used at the faces of nonhexahedral (or nonquadrilateral, in 2D) cells. The secondorder upwind scheme will also be used at partition boundaries when the parallel solver is used.

The QUICK scheme is available in the pressurebased solver and when solving additional scalar equations in the densitybased solver.

ThirdOrder MUSCL Scheme
This thirdorder convection scheme was conceived from the original MUSCL (Monotone UpstreamCentered Schemes for Conservation Laws) [ 378] by blending a central differencing scheme and secondorder upwind scheme as
(25.38) 
where is defined in Equation 25.35, and is computed using the secondorder upwind scheme as described in Section 25.3.1.
Unlike the QUICK scheme which is applicable to structured hex meshes only, the MUSCL scheme is applicable to arbitrary meshes. Compared to the secondorder upwind scheme, the thirdorder MUSCL has a potential to improve spatial accuracy for all types of meshes by reducing numerical diffusion, most significantly for complex threedimensional flows, and it is available for all transport equations.

The thirdorder MUSCL currently implemented in
FLUENT does not contain any fluxlimiter. As a result, it can produce undershoot and overshoot when the flowfield under consideration has discontinuities such as shock waves.


The MUSCL scheme is available in the pressurebased and densitybased solvers.

Modified HRIC Scheme
For simulations using the VOF multiphase model, upwind schemes are generally unsuitable for interface tracking because of their overly diffusive nature. Central differencing schemes, while generally able to retain the sharpness of the interface, are unbounded and often give unphysical results. In order to overcome these deficiencies, FLUENT uses a modified version of the High Resolution Interface Capturing (HRIC) scheme. The modified HRIC scheme is a composite NVD scheme that consists of a nonlinear blend of upwind and downwind differencing [ 257].
First, the normalized cell value of volume fraction, , is computed and is used to find the normalized face value, , as follows:
(25.39) 
where is the acceptor cell, is the donor cell, and is the upwind cell, and
(25.310) 
Here, if the upwind cell is not available (e.g., unstructured mesh), an extrapolated value is used for . Directly using this value of causes wrinkles in the interface, if the flow is parallel to the interface. So, FLUENT switches to ULTIMATE QUICKEST scheme (the onedimensional bounded version of the QUICK scheme [ 201]) based on the angle between the face normal and interface normal:
(25.311) 
This leads to a corrected version of the face volume fraction, :
(25.312) 
where
(25.313) 
and is a vector connecting cell centers adjacent to the face .
The face volume fraction is now obtained from the normalized value computed above as follows:
(25.314) 
The modified HRIC scheme provides improved accuracy for VOF calculations when compared to QUICK and secondorder schemes, and is less computationally expensive than the GeoReconstruct scheme.