Staggered grid

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Computational methodologies for forced convection
  1. One-Dimensional Steady-State Convection and Diffusion
    1. Central Difference Scheme
    2. Upwind Scheme
    3. Hybrid Scheme
    4. Exponential and Power Law Schemes
    5. A Generalized Expression of Discretization Schemes
  2. Multidimensional Convection and Diffusion Problems
  3. Numerical Solution of Flow Field
    1. Special Difficulties
    2. Staggered grid
    3. Pressure Correction Equation
    4. The SIMPLE Algorithm
  4. Numerical Simulation of Interfaces and Free Surfaces
  5. Application of Computational Methods
Staggered grid: (a) control volume for all other variables, (b) control volume for u, and (c) control volume for v.
Figure 1: Staggered grid: (a) control volume for all other variables, (b) control volume for u, and (c) control volume for v.

To overcome the checkerboard pressure field and develop an effective algorithm for the pressure field, one can use a staggered grid system (Patankar and Spalding, 1972; Patankar, 1980) that stores the pressure and all other variables on the main grid but calculates the velocity at the face of the control volume (see Fig. 1). The control volume for the velocity component in the x-direction, u, is staggered from the control volume for all other variables to the right direction by half a grid. Similarly, the control volume for v is staggered up by a half grid. The discretized equations for u and v will be obtained by integrating the momentum equation in their control volumes shown in Fig. 1 (b) and (c), respectively. The discretization equations in the staggered grid system for all variables except velocity are the same as those presented in the preceding subsection. Since velocity is defined at the face of the control volume, the grid Peclet numbers can be directly calculated from the velocity component, i.e., no assumption on the velocity profile between the grid points is needed. For the momentum equation in the x-direction, we will need to integrate eq. (4.287) for the control volumes of u shown in Fig. 1 (b). The result can be expressed as

a_{e}u_{e}=\sum{a_{nb}u_{nb}+b+A_{e}(p_{P}-p_{E})}

(1)

where the first term on the right hand side represents the summation of all the neighbor points of e. The effect of pressure has been separated from the source term, and the pressures at P and E were used to calculate the velocity ue. For a two-dimensional problem, the area on which the pressure difference acts on is Ae = Δy. Similarly, the discretization equation for v can be obtained by integrating eq.(4.287) for the control volume of vn shown in Fig. 1(c), i.e.,

a_{n}v_{n}=\sum{a_{nb}v_{nb}+b+A_{n}(p_{P}-p_{N})}

(2)

where the area on which pressure difference acts on is An = Δx for a two-dimensional problem. For a three-dimensional problem, eqs(4.292) and (4.293) are still valid except the neighbor points from the top and bottom should be included in the first term on the right-hand side. The areas on which the pressure difference act on should be modified to Ae = ΔyΔz and An = ΔxΔz. In addition to eqs. (4.292) and (4.293), another equation for the velocity component in the z-direction, wt, is also needed, i.e.,

a_{t}w_{t}=\sum{a_{nb}w_{nb}+b+A_{t}(p_{P}-p_{T})}

(3)

which is obtained by integrating the momentum equation in the z-direction for the control volume of wt that are staggered to the positive z-direction by a half grid. The area on which the pressure difference, pPpT, acts on is At = ΔxΔy.