6.5 Experiments and discussion
6.5.1 Flow around a circular cylinder
21]) using the nonequilibrium parts of the distribution function (see also [9]).
Instead of the integration method, another method is based on the momentum exchange originally proposed by Ladd [48] to compute the fluid force on a sphere suspended in the flow. This method is very easy to implement. The core of this method is to introduce the momentum transfer ψi∗(n,j) at the boundary nodexj (close to the obstacle) on the linkci at time level tn, which is defined by the difference between the distributions reaching and leaving this node,
ψi∗(n,j) =ci∗(fi(n+ 1,j) +fic∗(n,j)). (6.38) Then the boundary force acting on the cylinder can be approximated by
F˜b(n) = X
xj∈Γch
X
i∈Vj
ψi∗(n,j), (6.39)
where Vj is the collection of all the incoming directions at node j, Γch is the set of all boundary nodes in the neighborhood of the cylinder. Many numeri-cal experiments[52, 60] have been performed for both methods. Each demon-strates its advantage and weakness. However for the calculation of the force on a resting circular cylinder, both articles agree that the momentum-exchange method works quite well. The article [7] investigates the momentum trans-fer in connection with the second order boundary condition BFL. In [9] the accuracy of the above momentum-exchange method is analyzed, and a more accurate correction is proposed. Here we follow the suggestion to take use of the momentum-exchange method in the following numerical calculations.
Remark 13 Note that the lattice Boltzmann method (2.20) can approximate the solutions of Navier-Stokes equation (2.1) with 2nd order accurate velocity and 1st order accurate pressure generally. Therefore we can only expect the spatial derivatives of velocity to be 1st order accurate. The force evaluation (6.37)in the lattice Boltzmann method can only be1st order accurate too. Hence in order to obtain competitive values ofCd,Cl and△P to the reference provided in [70, 69, 27], the grid has to be fine enough.
6.5.1 Flow around a circular cylinder
The first two problems are two benchmark flows described detailedly in [70, 69, 27]. These articles also provide a number of reference values to verify the following results.
The inflow condition at inlet is defined by
U(0, y) =−4Umy(y−H)/H2, V(0, y) = 0, (6.40) where H = 0.41m is the height of channel. The diameter of the cylinder is D= 0.1m. The blockage ratioD/H is fixed for all the numerical experiments.
The fluid viscosity is ν = 10−3m2/s. The Reynolds number is defined by
Re = ¯U D/ν with a mean velocity ¯U = 2Um/3. These two test flows are distinguished by different values of Um.
(i)Um = 0.3m/s produces a Reynolds number Re = 20 and leads to a steady flow with a closed steady recirculation region consisting of two vortices behind the cylinder.
(ii)Um = 1.5m/syields a Reynolds number Re= 100 and renders an unsteady flow with a periodic vortex street appearing behind the cylinder.
Steady flow around a circular cylinder
Table 6.1 and 6.2 show our calculations of drag coefficient(Cd), lift coefficient(Cl), and the pressure difference(△P) for the steady flow. The top row provides the reference upper and lower bounds.
First let us observe the results (Table 6.1) in a short channel with fixedL/H= 2 on three grids respectively, it is found that finer grids render more accurate values in a relative short channel and agree well with the reference values. On the contrary table 6.2 shows the results on a fixed grid with largerL/H ∈ {3,4}. Since larger L/H values do not improve the results, we can conclude that all the outflow schemes have approached the real fluid movement to some extent.
Table 6.1: Comparison results for steady flow around a circular cylinder in a short channel.
L/H grid CD Cl △P
lower bound 5.5700 0.0104 0.1172
upper bound 5.5900 0.0110 0.1176
2 41×82 5.5792 0.0108 0.1194
NBC 2 82×164 5.5795 0.0117 0.1171
2 123×246 5.5816 0.0111 0.1174 2 41×82 5.5662 0.0108 0.1192
ZNS 2 82×164 5.5838 0.0117 0.1172
2 123×246 5.5713 0.01106 0.1172 2 41×82 5.5826 0.0111 0.1195
DNT 2 82×164 5.5816 0.0117 0.1172
2 123×246 5.5820 0.0111 0.1174
6.5.1 Flow around a circular cylinder 133
Table 6.2: Comparison results for steady flow around a circular cylinder in channels of varying length.
L/H grid CD Cl △P
lower bound 5.5700 0.0104 0.1172
upper bound 5.5900 0.0110 0.1176
2 41×82 5.5792 0.0108 0.1194
NBC 3 41×123 5.5793 0.0111 0.1194
4 41×164 5.5798 0.01105 0.1194 2 41×82 5.5662 0.0108 0.1192 ZNS 3 41×123 5.5805 0.0111 0.1194 4 41×164 5.5808 0.0111 0.1195 2 41×82 5.5826 0.0111 0.1195
DNT 3 41×123 5.5816 0.0117 0.1172
4 41×164 5.5811 0.0111 0.1195
Further, to compare the values of velocity and pressure carefully, we plot them along several cuts in the computational domain. The cut lines are chosen to meet different purposes (See the following figure 6.4). The dashed lines are three vertical cuts close to the front part of the cylinder (x= 0.15m), through the center (x = 0.2m) and at the back (x = 0.25m), which are used to show the velocity and pressure behavior around the cylinder. The dashdot lines are parallel to the solid walls and located next to the walls (y = 0.01m,0.40m) at a distance of 0.01m and through the center (y = 0.20m) of the cylinder respectively. They are used to observe the velocity and pressure varying along the x-coordinate. The solid lines atx= 2H,2H−0.01m,2H−0.02mare used to investigate the velocity and pressure varying versus the y-coordinate at outflow.
The test grids have the resolution 123×246 (L/H= 2).
Figure 6.4: The position of cutlines in the computational domain.
First let us see the results near the cylinder which are given in figure 6.5. All the calculations demonstrate a very reasonable and quite similar velocity and
pressure. This again verifies that all the outflow schemes can lead to the very close values for Cd,Cl and △P.
Next observe figure 6.6 which displays the results near the vertical linex= 2H (the solid lines in figure 6.4), where the outflow conditions are applied. From the plots of the horizontal velocity component (the first row in figure 6.6), we can see that all methods yield very close values. Near the symmetry axis of the channel, the results of NBC show a little deviation, whereas the results of ZNS and DNT coincide with each other. Besides, the plots of the vertical velocity component demonstrate that ZNS and DNT produce a boundary layer near the wall and NBC yields a smooth function with values close to zero.
In addition, the plots of pressure display again that ZNS and DNT behave similarly. Especially ZNS illustrates obvious boundary layers.
Now let us look at the plots along the dashdot cutlines (see figure 6.7). All the results agree with each other very well for the points a few grid sizes away from the right end at x = 2H. Moreover, before and behind the cylinder the curves are smooth. However, boundary layers appear near the outflow and they are stronger at points close to the wall. The reason may be that the boundary conditions at the corners are incompatible.
Unsteady flow around a circular cylinder
The Reynolds number for this unsteady flow is Re = 100. The numerical tests are carried out on three grids 82×164(L/H = 2), 82×246(L/H = 3), 82×410(L/H = 5).
Figure 6.10 plots the quantities Cd, Cl and △P for 1000 time steps after the initial transient behavior. All the results show that the values are getting closer to the reference value as the channel becomes longer, and ZNS and DNT produces much better results than NBC on all grids. Comparing the size of these quantities carefully, we find that ZNS and DNT yield Cdand△P for the short channel (L/H = 2) closer to the reference value than the ones given by NBC on a longer channel (L/H = 3). A similar comparison also holds for the channel L/H = 3 with ZNS or DNT andL/H = 5 with NBC. Comparatively, DNT and ZNS influences the inner flow most weakly, so that the short channel(L/H = 3) already produces nearly the same values as the one with large L/H = 5.
Another test of the effect of the outflow condition is to investigate the vortex street developed behind the cylinder. To check how this structure is influenced by the outflow condition, we compare velocity and pressure results along the central line at the same time level but for various choices of the channel length.
The results are shown in the interval reflecting the shortest channel (see Fig.
6.11, Fig. 6.12 and Fig. 6.13 for the output of the various methods). From Fig. 6.11 we see that all quantities behave differently behind the cylinder with respect to differentL/Hratios. It demonstrates that the NBC outflow condition (6.13) has a comparably strong impact on the inner flow. On the contrary, ZNS and DNT yield more similar variations of these quantities. Only a slight phase difference occurs.
6.5.1 Flow around a circular cylinder 135
Figure 6.5: Results for the steady flow. The rows represent the horizontal velocity component, the vertical component and the pressure respectively, while the columns represent the three cut lines atx= 0.15m(first column), x= 0.2m(middle column), 0.25m (right column). In each plot the methods are distinguished by: ZNS (∗), NBC (∇) and DNT (◦).
0 50 100
Figure 6.6: Results for the steady flow along three cut lines x= 2H (left column), x= 2H −0.01m (middle column), x = 2H −0.02m (right column). The horizontal velocity (1st row), the vertical velocity (2nd row) and the pressure (3rd row) are shown for three methods ZNS (∗), NBC (∇) and DNT (◦).
6.5.1 Flow around a circular cylinder 137
Figure 6.7: The plots of velocity and pressure along three cut lines parallel to the wall for the steady flow: y = 0.01m(left column), y= 0.20m(middle column), y= 0.40m (right column). The horizontal velocity (1st row), the vertical velocity (2nd row) and the pressure (3rd row) are shown for three methods ZNS (∗), NBC (∇) and DNT (◦).
1.92 1.94 1.96 1.98 2
Figure 6.8: The plots ofCd(left),Cl(middle) and△P(right) for the unsteady flow with NBC outflow condition. Solid line stands for the result in the channel withL/H = 5, dashed line for the result with L/H= 3 and dotted line for the result withL/H = 2.
The dash-dotted line is the upper and lower bound for the amplitude of these quantities.
1.92 1.94 1.96 1.98 2
Figure 6.9: The plots ofCd(left),Cl(middle) and△P(right) for the unsteady flow with ZNS outflow condition. Solid line stands for the result in the channel with L/H = 5 grids, dashed line for the result with L/H = 3 and dotted line for the result with L/H = 2. The dash-dotted line is the upper and lower bound for the amplitude of these quantities.
Figure 6.10: The plots ofCd(left), Cl(middle) and △P(right) for the unsteady flow with DNT outflow condition. Solid line stands for the result in the channel with L/H = 5 grids, dashed line for the result withL/H= 3 and dotted line for the result with L/H= 2. The dash-dotted line is the upper and lower bound for the amplitude of these quantities.
6.5.1 Flow around a circular cylinder 139
Figure 6.11: The velocity components (left: horizontal, middle: vertical) and pressure (right) along the central line for the unsteady flow with NBC outflow condition. +,◦ and∇stand for the result in the channel withL/H= 2,3,5 respectively.
0 50 100 150
Figure 6.12: The velocity components (left: horizontal, middle: vertical) and pressure (right) along the central line for the unsteady flow with ZNS outflow condition. +,◦ and∇stand for the result in the channel withL/H= 2,3,5 respectively.
0 50 100 150
Figure 6.13: The velocity components (left: horizontal, middle: vertical) and pressure (right) along the central line for the unsteady flow with DNT outflow condition. +,◦ and∇stand for the result in the channel withL/H= 2,3,5 respectively.
50 100 150 50 100 150 50 100 150
50 100 150 200 50 100 150 200 50 100 150 200
50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400
Figure 6.14: The pressure isolines for three outflow conditions (NBC (left), ZNS (middle), DNT (right)) in channels of different length. Magnified versions can be found in appendix A.
Further, in order to observe the behavior of velocity and pressure in the whole domain, we show their contour or isoline plots in the computational domain.
From Fig. 6.14 which is the plots for the pressure, we see that the lattice Boltzmann method with DNT and ZNS can result in a clear vortex street for the channels of different length. However, the periodicity of the flow is not so clear with NBC near the outflow, in particular, this phenomenon is more obvious in the case with short channel. The same situation occurs also in the case of velocity. See the appendix A in which all the figures are collected.