3 Basics of numerical modelling
3.4 Finite Volume Method
3.4.2 Discretisation in space
3.4.2.1 Numerical grids
To solve the basic flow equations the flow domain is divided in single control volumes Ωi. A knot in its core defines each control volume. The choice of the numerical grid, which describes the flow domain, is of particular importance for the precision of the calculations. Basically it can be distinguished between structured and unstructured grids. (Figure 3.1)
Structured grids are orthogonal grids which means that the meshes are nearly vertical to each other’s. For equidistant grids the distances between the single meshes are equal.
The nodes in unstructured grids are distributed arbitrarily. Each point is connected to several neighbour points. Thereby faces are generated who feature the shape of triangles or alike. This shape allows a better adaptation of the numerical grid to complicated geometries. The high resolution of the unstructured grids leads to a higher computing time because the flow equations have to be solved more often for the same flow domain.
Finite Volume Method 35
Figure 3.1: Unstructured - structured numerical grids [Valentin & Rinaldi, 2001]
Local refinements are necessary at important places of the flow domain to get more detailed results. The transition between the original grids the refined one should not be abrupt. To ensure a stable calculation the difference in size between two neighbouring cells should not exceed the factor two.
Beside the static grids, which keep their shape during the calculation, there are adaptive grids to illustrate the flow domain. These grids can change their shape and adjust themselves to the water level. They do not move along the river for example like the third kind of grids the moveable grids. They can describe the forward movement of a wave along a dry bottom. More information’s on numerical grids can be found for example in Buergisser (1998) and Oertel and Laurien (2003).
3.4.2.2 Divergence form
The discretisation in space is carried out by the definition of the basic hydro-dynamic equations in divergence form.
∂f
∂t +div Φf =qf (3.27)
The terms of the three-dimensional incompressible Navier-Stokes equation are shown in table 3.2.
Equation f Φf qf
x-momentum u1 u1ui−ν grad u1 −1ρ∂x∂p1 −g1 y-momentum u2 u2ui−ν grad u2 −1ρ∂x∂p2 −g2 z-momentum u3 u3ui−ν grad u3 −1ρ∂x∂p3 −g3
Continuity ρ ρ ui 0
Table 3.2: Terms of the divergence form [Malcherek, 2001a]
3.4.2.3 Integral form of the basic flow equations
The divergence form of the basic flow equations are integrated across the control volumes Ωi. Then they are transformed by the Gau integral theorem. (Equation 3.28) The usage
of the Gauss integral theorem reduces the volume integral to a boundary integral by leaving out the divergence.
Ωi
∂f
∂t dΩ I
+
S
Φf dS II
=
Ωi
qf dΩ III
(3.28)
I: Integration of time
II: Flux term: Summation of all fluxes across the boundaries
III: Source term: Energising forces of the flow, for example the pressure gradient This equation states that each temporal change of the valuef in the volume Ωiis defined by the balance of the fluxes Φi across the boundaries and through the sources and sinks.
Derivation terms of the basic flow equations drop out because of the integration of the divergence term. The second order derivations of the diffusion terms are reduced to first order derivations. This makes the differential equations easier to solve.
3.4.2.4 Treatment of the single terms
Integration of time
Ωi
∂f
∂t dΩ = ∂
∂t
Ωi
f dΩ ∂
∂t(Vi fi)Vifin+1−fin
∆t (3.29)
The integration of the derivation of time is calculated approximately by the volume Vi of the control volume Ωi.
Source term
The integration across the finite volumes Ωi are substituted by a product with Vi.
Ωi
qf dΩVi qfi (3.30)
Flux term
The in- and out-flows are summed up across the according boundaries with the length of their edges. (Figure 3.2)
S
Φf dS −Φi−1/2·∆y+ Φi+1/2·∆y−Φj−1/2·∆x+ Φj+1/2 ·∆x (3.31)
A negative algebraic sign in equation 3.31 means a flow into the grid cell. A positive algebraic sign stands for an outflow of the grid cell. The interpolation of the fluxes across
Finite Volume Method 37 the respective borders has to be done with the known values from nodes of the grid cells.
[Valentin & Rinaldi, 2001]
∆x
∆y
j + 1
j
j - 1
i - 1 i i + 1
∆x
∆y
j + 1
j
j - 1
i - 1 i i + 1
Figure 3.2: Definition of nodes for the flux term [Valentin & Rinaldi, 2001]
3.4.2.5 Interpolation methods
The calculation of the fluxes across the cell boundaries is done using interpolations with the known values of the cell nodes. Therefore numerous methods are available. Gener-ally it can be divided between first, second or higher order methods.
First order methods generate discrete equations, which are easy to solve but mostly the results are not very precise. Steep gradients are not reproduced sharp, they smear because of numerical diffusion. One solution to overcome this problem is to refine the numerical grid but this leads to much longer calculation times. Example for first order methods are Centrale methods, Upstream methods or Upwind difference methods.
Due to their higher accuracy second or higher order methods can reproduce steep gradi-ents better, but they also more fragile to numerical instabilities. In this case a refinement of the numerical grid can lead to stable calculation but also to longer computational times. Examples for high order methods are the Quick method, the Centrale difference method and the Linear upwind differencing method. Further methods of interpolation are documented in the StarCD Methology (2002).
One alternative to avoid the interpolation is the usage of so called staggered grids. They define nodes for each physical value at favourable locations and control volumes for each basic equation. [Valentin & Rinaldi, 2001]
The calculations in the following chapters were carried out using the First Order Up-wind Differencing Scheme in the numerical model StarCD. The reason for choosing a first order method was the size of the flow domain which had to be reproduced with a numerical grid. The investigated reservoir sewer has a length of 336 m and a width of 3.2 m which leads, even for a first order scheme, to a large number of grid cells and long calculation times. To use a high order interpolation method it would have been necessary to increase the number of grid cell to get stable results. But the main focus of the investigations was based on the calculation of the bottom shear stress and the cleaning effect of the flush wave for long distances. It was not the target to reproduce the breaking wave in its initial phase precisely. Therefore the time saving first order
scheme was chosen. In an additional investigation the initial phase was modelled with a lengthwise short but refined grid to reproduce the breaking wave.
The First Order Upwind Differencing Scheme in the model StarCD is based on different assumptions to the before mentioned ones. The flux term of equation 3.28 is split into a convective (Cj) and a diffusive term (Dj). The conservation of mass is assured in this method but a too large iteration time stepδt can lead to numerical diffusion.
The diffusive ratio is described with equation 3.32.
Dj ≈ΓΦ,j[fj(ΦN −ΦP) +{grad Φ·S−fj gradΦ·dNP}] (3.32) The first summand in equation 3.32 describes the common diffusion between the points P and N. The second summand in the cambered brackets involves the cross-diffusion.
[Star-CD Version 3.15, 2002]
The convective ration is characterized as the product of the mass fluxFj across the cell boundaries and the mean value Φj at the border faces of the cells. (Equation 3.33)
Cj ≡Fj ·Φj (3.33)
The mass flux is defined as follows
Fj ≡(ρ· ur·S). (3.34)
Equation 3.35 applies for the First Order Upwind Differencing Scheme.
Cj =Fj ·
ΦP for Fj ≥0
ΦN+ for Fj <0 (3.35)
The definition of the adjoining cells and their nodes is displayed in figure 3.3.
N ++
P N + N -- N
-F3 Face j
N ++
P N + N -- N
-F3 Face j
Figure 3.3: Definition of cells and nodes First Order Upwind Differencing Scheme [Star-CD Version 3.15, 2002]
Finite Volume Method 39