Control

In order to satisfy the third requirement of the previous section, a number of short experiments have been performed to determine suitable choices for the polynomial orderP and element

meshK;r. Suitable in this sense means that the parameters are chosen which are small

enough to minimise computational effort, whilst providing a sufficiently resolved domain up to Re3;000and hence provide the ability to simulate fully-turbulent flows.

In these simulations, the domain length and axial resolution are fixed atL _D 10D and Nx D386respectively, providing a domain capable of generating data over long timescales

in short periods of execution time, and also allowingto remain sufficiently small. Periodic

boundary conditions are taken in the axial direction and the flow is driven using the volumetric flux technique of section 3.2. A fully turbulent flow is simulated for a short period of time using a very high-resolution mesh. The resulting flow field provides an initial condition to test any given candidate meshKr;.

A spectral interpolation technique is used to project the high-order flow field onto each candidate mesh, and as a result it is likely that some small numerical errors may appear. For instance, some meshes may contain nodal positions near to or on the boundary which do not appear on the high-resolution mesh. In order to alleviate these problems, the field is allowed to ‘burn in’ for 100-200 time units, providing sufficient time for transient effects resulting from the interpolation to decay.

After the burn-in period, a large number of quantities are recorded; of particular interest are:

regular outputs of field data;

0 50 100 150 200 t 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 uz

Figure 4.2:Trace of the axial velocityu.0; t /for the instigation of turbulence in a pipe of

lengthL _D20D. The figure is split up into three regions: (a) initial deformation of the

turbulent profile; (b) onset of instabilty; (c) transition to statistically steady turbulent flow.

the forcing term˛.t /used to impose the constant volumetric flux condition;

shear stress data around the boundary to determinew.

By comparing these quantities to existing literature, or the results from the high-resolution mesh, we can draw conclusions pertaining to the suitability of any given candidate mesh. The choice of high-resolution mesh is critical, since it is the benchmark to which all other meshes are compared. Initial testing revealed that the reflective mesh of figure 4.1(b) using

P _D 12 provides a sufficient mesh to provide a good comparison. Indeed, comparison

against literature (in particular McIveret al., 2000, which also uses this mesh) shows it remains well-resolved up to Re5;000.

To generate the initial turbulent field, we use a standard technique seen in much of the existing literature. An initial laminar conditionu0is generated with the mean profile of a

turbulent field. To this, we add normally-distributed white noise with a small amplitude. A good approximation to the mean profile can be obtained by using a one-seventh power law, so that the initial condition becomes

u0.x; r; /D.C.R r/

1

8 9 10 11 12 13 P 0:0 0:2 0:4 0:6 0:8 1:0 1:2 1:4 ku+_z u_Q+_zk1 Reflective mesh K6;5mesh K8;4mesh K8;5mesh

Figure 4.3:L1error of the near-wall axial velocityu+zfor various polynomial ordersP for

four candidate meshes.

whereC is a constant ensuring that the fluxQ.u0/is approximately equal to the bulk velocity

andn.x; r; /is normally distributed noise with zero mean. Figure 4.2 shows the resulting

transition to turbulence for this initial condition in a pipe of lengthL_D20Dand diameter D D 1. Part (a) of the figure shows a deformation of the initial mean-turbulent profile

into one approaching laminarity. However, after around 75 time units, part (b) shows the sudden growth of the small perturbations, leading to the initial onset of turbulent flow. These perturbations grow most rapidly in the axial component of velocity, and as such the flowrate restriction forces the mean velocity to remain near its maximum ofu_maxD2. After a short

period of apparent stability, the disturbance grows large enough in the transverse velocity that a sharp drop in mean velocity to a statistically steady turbulent state is seen, as shown in part (c) of the figure.

One remaining question is that of the length of time required to generate suitable averages for the field data and wall-shear stress. Here, each simulation of a candidate mesh gathers precisely5102 time units of data after burn-in, a quantity which is regarded to be far

more than is statistically necessary. For the high-resolution mesh, we use a field obtained by averaging over2103time units to obtain a more accurate temporal average.

logarithmic plot of the non-dimensionalised mean axial wall velocity u+ as a function

of wall unitsy+. This curve has a distinctive shape and is very sensitive to numerical error.

To obtain a general overview as to the suitability of each mesh, a velocity fielduQ+.y+/is

generated and theL1errork Qu+ u+k1is calculated, whereu+is the field obtained from

a simulation on the high-resolution mesh. Figure 4.3 shows these errors for polynomial orders76P 612on the reflective mesh of figure 4.1(b) and a variety of rotational meshes:

namely,K6;5andK8;r forrD4; 5.

There are a number of immediate conclusions that can be drawn from this plot. Firstly, increasingP broadly decreases the error for all meshes, demonstrating that this is a rea-

sonably suitable metric by which to compare them. In addition, the reflective mesh (for whichP _D12is the high-resolution comparison mesh) shows consistently similar results

forP >10, indicating its suitability for comparison atP _D12.

The worst performing mesh is clearlyK6;5, which shows persistent under-resolution even

at high polynomial order. The radial spacings here are precisely those used in the K8;5

mesh which performs significantly better at all polynomial orders. This indicates that 6 azimuthal elements are insufficient to correctly resolve the domain, and we therefore require

>8azimuthally placed elements. To determine the suitability of the radial resolution, we

consider another meshK8;4 which has fewer radial elements. AtP D7and8, the mesh

performs similarly to theK6;5mesh, and indicates the domain is significantly under-resolved

in the radial direction. However, forP >10, the error is more similar to that seen in theK8;5

mesh indicating that the errors seen are more likely to arise from azimuthal under-resolution. These results demonstrate that theK8;4mesh atP D12provides an ideal compromise of

azimuthal and radial resolution, providing a very close approximation to the comparison mesh. Figure 4.4 shows a standard semi-logarithmic plot ofu+against wall unitsy+for the

comparison mesh (solid blue line) and theK8;4(dots) mesh which demonstrate the accuracy

of the fit across the whole of the curve.

The significant advantage of this mesh is the improvement in computational speed. For any mesh, the total number of degrees of freedom in the system is given byNdofD4NxNelP2.

(The multiplication by four arrives from the three components of the velocity field and the pressure field.) For the high resolution mesh,Nel D64, whereas theK8;4mesh only has

100 ₁₀1 ₁₀2 y+ 0 5 10 15 20 u+_z High-order mesh,P _D12 K4;8mesh,P D12

Figure 4.4:Axial component of the near-wall velocityu+_zin a full-turbulence simulation at

ReD3000using two different types of meshes.

around two at any givenP.