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25.3.1 Spatial Discretization

By default, FLUENT stores discrete values of the scalar $\phi$ at the cell centers ( $c_0$ and $c_1$ in Figure  25.2.1). However, face values $\phi_f$ are required for the convection terms in Equation  25.2-2 and must be interpolated from the cell center values. This is accomplished using an upwind scheme.

Upwinding means that the face value $\phi_f$ is derived from quantities in the cell upstream, or "upwind,'' relative to the direction of the normal velocity $v_n$ in Equation  25.2-2. FLUENT allows you to choose from several upwind schemes: first-order upwind, second-order upwind, power law, and QUICK. These schemes are described in Sections  25.3.1- 25.3.1.

The diffusion terms in Equation  25.2-2 are central-differenced and are always second-order accurate.

First-Order Upwind Scheme

When first-order accuracy is desired, quantities at cell faces are determined by assuming that the cell-center values of any field variable represent a cell-average value and hold throughout the entire cell; the face quantities are identical to the cell quantities. Thus when first-order upwinding is selected, the face value $\phi_f$ is set equal to the cell-center value of $\phi$ in the upstream cell.


First-order upwind is available in the pressure-based and density-based solvers.

Power-Law Scheme

The power-law discretization scheme interpolates the face value of a variable, $\phi$, using the exact solution to a one-dimensional convection-diffusion equation

 \frac{\partial}{\partial x} (\rho u \phi) = \frac{\partial}{\partial x} {\Gamma} \frac{\partial \phi}{\partial x} (25.3-1)

where ${\Gamma}$ and $\rho u$ are constant across the interval $\partial x$. Equation  25.3-1 can be integrated to yield the following solution describing how $\phi$ varies with $x$:

 \frac{\phi (x) - \phi_0}{\phi_L - \phi_0} = \frac{\exp ({\rm Pe} \frac{x}{L}) -1}{\exp ({\rm Pe}) -1} (25.3-2)


$\phi_0$ = $\phi\vert _{x=0}$
$\phi_L$ = $\phi\vert _{x=L}$

and Pe is the Peclet number:

 {\rm Pe} = \frac{\rho u L}{{\Gamma}} (25.3-3)

The variation of $\phi(x)$ between $x=0$ and $x=L$ 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 $\phi$ at $x=L/2$ 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 first-order scheme for FLUENT.

Figure 25.3.1: Variation of a Variable $\phi$ Between $x=0$ and $x=L$ (Equation  25.3-1)

If the power-law scheme is selected, FLUENT uses Equation  25.3-2 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 $\phi$ at $x=L/2$ is approximately equal to the upstream value. When Pe=0 (no flow, or pure diffusion), Figure  25.3.1 shows that $\phi$ may be interpolated using a simple linear average between the values at $x=0$ and $x=L$. When the Peclet number has an intermediate value, the interpolated value for $\phi$ at $x=L/2$ must be derived by applying the "power law'' equivalent of Equation  25.3-2.


The power-law scheme is available in the pressure-based solver and when solving additional scalar equations in the density-based solver.

Second-Order Upwind Scheme

When second-order accuracy is desired, quantities at cell faces are computed using a multidimensional linear reconstruction approach [ 21]. In this approach, higher-order accuracy is achieved at cell faces through a Taylor series expansion of the cell-centered solution about the cell centroid. Thus when second-order upwinding is selected, the face value $\phi_f$ is computed using the following expression:

 \phi_{f,SOU} = \phi + \nabla \phi \cdot {\vec r} (25.3-4)

where $\phi$ and $\nabla \phi$ are the cell-centered value and its gradient in the upstream cell, and $ {\vec r}$ is the displacement vector from the upstream cell centroid to the face centroid. This formulation requires the determination of the gradient $\nabla \phi$ in each cell, as discussed in Section  25.3.3. Finally, the gradient $\nabla \phi$ is limited so that no new maxima or minima are introduced.


Second-order upwind is available in the pressure-based and density-based solvers.

Central-Differencing Scheme

A second-order-accurate central-differencing 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 central-differencing scheme calculates the face value for a variable ( $\phi_f$) as follows:

 \phi_{f,{\rm CD}} = \frac{1}{2} \left(\phi_0 + \phi_1 \right... ..._{0} \cdot \vec{r}_0 + \nabla \phi_{1} \cdot \vec{r}_1 \right) (25.3-5)

where the indices 0 and 1 refer to the cells that share face $f$, $\nabla \phi_{r,0}$ and $\nabla \phi_{r,1}$ are the reconstructed gradients at cells 0 and 1, respectively, and $\vec{r}$ is the vector directed from the cell centroid toward the face centroid.

It is well known that central-differencing schemes can produce unbounded solutions and non-physical 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 central-differencing scheme. In this approach, the face value is calculated as follows:

$\displaystyle \phi_f = \underbrace{\phi_{f,{\rm UP}}}_{\mbox{implicit part}} + ... ...\; \underbrace{( \phi_{f,{\rm CD}} - \phi_{f,{\rm UP}})}_{\mbox{explicit part}}$     (25.3-6)

where UP stands for upwind. As indicated, the upwind part is treated implicitly while the difference between the central-difference and upwind values is treated explicitly. Provided that the numerical solution converges, this approach leads to pure second-order differencing.


The central differencing scheme is available only in the pressure-based 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 subgrid-scale 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 NVD-scheme that consists of a pure central differencing, a blended scheme of the central differencing and the second-order upwind scheme, and the first-order upwind scheme. It should be noted that the first-order 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 pressure-based 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 higher-order value of the convected variable $\phi$ at a face. QUICK-type schemes [ 202] are based on a weighted average of second-order-upwind and central interpolations of the variable. For the face $e$ in Figure  25.3.2, if the flow is from left to right, such a value can be written as

 \phi_e = \theta\left[{S_d \over S_c + S_d} \phi_P + {S_c \o... ...\over S_u + S_c} \phi_P - {S_c \over S_u + S_c} \phi_W\right] (25.3-7)

Figure 25.3.2: One-Dimensional Control Volume

$\theta=1$ in the above equation results in a central second-order interpolation while $\theta=0$ yields a second-order upwind value. The traditional QUICK scheme is obtained by setting $\theta = {1/8}$. The implementation in FLUENT uses a variable, solution-dependent value of $\theta$, 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 second-order upwind discretization scheme (described in Section  25.3.1) will be used at the faces of non-hexahedral (or non-quadrilateral, in 2D) cells. The second-order upwind scheme will also be used at partition boundaries when the parallel solver is used.


The QUICK scheme is available in the pressure-based solver and when solving additional scalar equations in the density-based solver.

Third-Order MUSCL Scheme

This third-order convection scheme was conceived from the original MUSCL (Monotone Upstream-Centered Schemes for Conservation Laws) [ 378] by blending a central differencing scheme and second-order upwind scheme as

 \phi_{f} = \theta {\phi_{f,{\rm CD}}} + (1-\theta) \phi_{f,{\rm SOU}} (25.3-8)

where ${\phi_{f,{\rm CD}}}$ is defined in Equation  25.3-5, and $\phi_{f,{\rm SOU}}$ is computed using the second-order 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 second-order upwind scheme, the third-order MUSCL has a potential to improve spatial accuracy for all types of meshes by reducing numerical diffusion, most significantly for complex three-dimensional flows, and it is available for all transport equations.


The third-order MUSCL currently implemented in FLUENT does not contain any flux-limiter. As a result, it can produce undershoot and overshoot when the flow-field under consideration has discontinuities such as shock waves.


The MUSCL scheme is available in the pressure-based and density-based 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 non-linear blend of upwind and downwind differencing [ 257].

First, the normalized cell value of volume fraction, $\tilde{\phi_c}$, is computed and is used to find the normalized face value, $\tilde{\phi}_f$, as follows:

 \tilde{\phi_c} = \frac{\phi_D - \phi_U}{\phi_A - \phi_U} (25.3-9)

Figure 25.3.3: Cell Representation for Modified HRIC Scheme

where $A$ is the acceptor cell, $D$ is the donor cell, and $U$ is the upwind cell, and

 \tilde{\phi}_f = \left\{ \begin{array}{ll} \tilde{\phi_c} & ... ... 1 & \mbox{$0.5 \leq \tilde{\phi_c} \le 1$} \end{array}\right. (25.3-10)

Here, if the upwind cell is not available (e.g., unstructured mesh), an extrapolated value is used for $\phi_U$. Directly using this value of $\tilde{\phi}_f$ causes wrinkles in the interface, if the flow is parallel to the interface. So, FLUENT switches to ULTIMATE QUICKEST scheme (the one-dimensional bounded version of the QUICK scheme [ 201]) based on the angle between the face normal and interface normal:

 \tilde{\phi^{UQ}_f} = \left\{ \begin{array}{ll} \tilde{\phi_... ...t) & \mbox{$0.5 \leq \tilde{\phi_c} \le 1$} \end{array}\right. (25.3-11)

This leads to a corrected version of the face volume fraction, $\tilde{\phi}_f^*$:

 \tilde{\phi}_f^* = \tilde{\phi}_f \sqrt{\cos{\theta}} + (1-\sqrt{\cos{\theta}}) \tilde{\phi^{UQ}_f} (25.3-12)


 \cos{\theta} = \frac{\nabla \phi \cdot \vec{\bf {d}}}{ \vert \nabla \phi \vert \vert \vec{\bf {d}}\vert } (25.3-13)

and $\vec{\bf {d}}$ is a vector connecting cell centers adjacent to the face $f$.

The face volume fraction is now obtained from the normalized value computed above as follows:

 \phi_f = \tilde{\phi}_f^* (\phi_A - \phi_U) + \phi_U (25.3-14)

The modified HRIC scheme provides improved accuracy for VOF calculations when compared to QUICK and second-order schemes, and is less computationally expensive than the Geo-Reconstruct scheme.

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