31.8 Two-Dimensional Problems of First Order in Time and Second-Order in Space

31.8.1 Introduction

The next case we want to study are differential equations, which are first order in time and second order in space. Obviously, the time-dependent Navier–Stokes equation for Poiseuille flow is a typical example of such an equation (see Eq. 18.1). It is given in analogy to Eq. 31.39 by

∂g∂txyt+∂Fx∂xxyt+∂Fy∂yxyt=0

si146_e

where we restrict ourselves again to the homogeneous case. As stated, the inhomogeneous case only brings in additional terms on the right-hand side that must be rewritten and integrated as detailed in section 31.3.3. In this case, the vector function F now contains partial derivatives as well. It is given by

F→=∂g∂x∂g∂y

si147_e

31.8.2 Flux Approximations

The derivation of this case is exactly identical to the derivation we made in section 31.7. The only adjustment we need to make is the approximation to the changes Fxy,1→2jk,Fxy,2→3jk,Fxy,3→4jk si148_e, and Fxy,4→1jk si149_e. We need to make this adjustment, as we require the evaluation of the first derivatives, i.e., of the fluxes instead of the actual values. Luckily, this is easy and we already have a suitable approximation function. As we assume linear changes within the cells, we can reuse the approximation to the fluxes on the boundaries we made in Eq. 31.16. Using this equation, we find the following approximations

Fx,1→2jk≈Fxjk+Fxj,k−1ΔxjkFx,2→3jk≈Fxj+1,k−FxjkΔxjkFx,3→4jk≈Fxj,k+1−FxjkΔxjkFx,4→1jk≈Fxjk−Fxj−1,kΔxjkFy,1→2jk≈Fyjk−Fyj,k−1ΔyjkFy,2→3jk≈Fyj+1,k−FyjkΔyjkFy,3→4jk≈Fyj,k+1−FyjkΔyjkFy,4→1jk≈Fyjk−Fyj−1,kΔyjk

si151_e

Please note that the division by Δx(j,k) and Δy(j,k) is due to the reconversion from the local coordinate system within the cell to the global coordinates. See section 31.5.5 for details on this procedure.

Having derived an approximation for the fluxes, we can now assemble the numerical scheme.

31.8.3 Regular Axes-lncident Grid

We will again restrict our derivation on regular grids that are incident with the coordinate axes. We can then reuse Eq. 31.45 and simply insert the flux approximations instead of the function approximations we used earlier. We then find

∫Ω∂Fx∂x+∂Fy∂ydΩ=−Fy,1→2−Fy,3→4Δxjk+Fx,2→3−Fx,4→1Δyjk=−Fyjk−Fy(j,k−1)−Fyj,k+1+FyjkΔxjkΔyjk+Fxj+1,k−Fxjk−Fxjk+Fxj−1,kΔyjkΔxjk=Fyj,k+1−2Fyjk+Fyj,k−1ΔxjkΔyjk+Fxj+1,k−2Fxjk+Fxj−1,kΔyjkΔxjk

si152_e  (Eq. 31.48)

31.8.4 Numerical Scheme

We are almost done, as it only remains to set in Eq. 31.48 into Eq. 31.40 to derive our numerical scheme. Doing this we find

ddt∫ΩgdΩ+Fyj,k+1−2Fyjk+Fyj,k−1ΔxjkΔyjk+Fxj+1,k−2Fxjk+Fxj−1,kΔyjkΔxjk=0dg¯jkdtΔxjkΔyjk+Fyj,k+1−2FyJK+Fyj,k−1ΔxjkΔyjk+Fxj+1,k−2Fxjk+Fxj−1,kΔyjkΔxjk=0dg¯dtjk+Fyj,k+1−2Fyjk+Fyj,k−1Δyjk2+Fxj+1,k−2Fxjk+Fxj−1,kΔxjk2=0

si153_e  (Eq. 31.49)

Eq. 31.49 is the numerical scheme sought. Again, it is a first-order ODE in time. The spatial terms have been replaced with equations, which are effectively central schemes for deriving the second derivative that we already know from the Taylor series (see Eq. 4.10). The same formula would have been derived when using an FDM scheme.

We will not implement this numerical scheme at this point, as we will return to this equation in section 33.2. There we derive the same equation using a symmetrical grid with Δx(j,k) = Δy(j,k). The equation we will use is Eq. 33.3. If you compare Eq. 31.49 and Eq. 33.3, you will see that they are identical.

31.8.5 Comment on the Grid

Obviously, one could ask the question why should FVM be preferred if it yields the same equations as a FDM scheme? The fact that we derived the same equation is only due to the nature of the grid we chose. As stated, FDM works on regular grids. It cannot be implement in non-ordered grids. However, FVM can operate on any grid, even when the boundaries of the cells are not incident with the coordinate axes. In this case, we would obviously derive different equations.

31.9 Summary

In this chapter we have introduced FVM, which is one of the most commonly used methods in fluid mechanics. In contrast to FDM, the concept of FVM is not to estimate the differential of the dependent function. FVM performs a conservation over each cell in the computation domain. In order to do so, the differential equation is integrated and the resulting equation rewritten using Gauss’s theorem. The equation will then depend on the value of the fluxes on the boundary of each cell. By choosing suitable approximation functions, these flux values can be calculated and balanced for each cell. As a result, the local value of the solution can be obtained and thus reconstructed over the whole domain.

We have studied an example using FVM to solve the one-dimensional heat equation using a spreadsheet analysis tool, e.g., Microsoft Excel. As we have seen, this implementation of FVM yielded a close-to-exact solution. We also discussed differential equations with first- and second-order spatial derivatives that we commonly encounter in fluid mechanics. As we have seen, they can be treated conveniently by FVM, yielding equations that can be directly implemented as a numerical scheme.

FVM is a useful tool in many areas of computational physics. The method is very stable and even handles solutions with discontinuities if the domain is suitably structured. Compared to FDM, we do not require regular grids for FVM, which makes this method extremely suitable for curved geometries.

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