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Microscale High Resolution Meteorological Model. Tests of Used Methods and the First Preliminary Results.
Josef Brechler, Vladimír Fuka
Department of Meteorology and Environment Protection, Faculty of Mathematics and Physics, Charles University, V Holesovickach 2, 180 00 Prague 8, Czech Republic
Phone: (+420) 22191 2549, fax: (+420) 22191 2533, e-mail: firstname.lastname@example.org
Abstract: A new high resolution model of flow within complicated (complex) geometry is developed and its basis is described in this contribution. The model uses finite volume approach and the high resolution Godunov methods of solution have been applied. At this moment the model describes 2D flow mainly because of the amount of processor time decrease but the transition to the full 3D is straightforward. The complicated geometry is implemented via so called immersed boundary method (ImBM) that enables to use orthogonal Carthesian grid. At this moment the flow is supposed to be laminar and neutrally stratified, the appropriate turbulence model and temperature stratification will be applied later. In this article results of tests of the used numerical methods are discussed together with comparison of the presented model preliminary results with the reference ones. These tests were performed for validation of used methods and standards Sod problem has been applied. The reference data sets dealt with the upper lit driven cavity flow and the model results were compared also with the analytical Blasius solution of the boundary layer development above a flat plate. Then the 2D flow around the square oriented in a various ways with respect to it has been performed and results are shown.
Key-words: flow in complicated geometry, finite volume method, Godunov´s method, immersed boundary.
To know information about the flow pattern in geometrically complicated areas like the urban ones is sometimes a highly demanded task. For example if the dispersion of the traffic air-pollution is needed to be known in some part of the city the appropriate model tool with corresponding resolution is to be used because monitoring network is not usually dense enough to describe this situation well. Similar case deals with the industrial accidents accompanied with releases of dangerous gaseous pollution into the atmosphere inside the settled areas or close to them. The “classical” Gaussian dispersion models are non-usable in such situations as the flow field deformation on the street level together with the local circulation systems result in very complicated flow conditions (see, for example, Nakayama H. and Tamura T., 2004). A possible approach for a construction of the flow field consists in using of detailed enough models - so-called CFD (Computational Fluid Dynamics) models that can describe all the desired details of the complicated air-flow pattern better.
Commercial CFD models like Star-CD, Fluent and the others are sometimes used for environmental problems as diagnosis or prognosis of air-flow pattern or air-pollution dispersion fields. But these models have to cover a wide spectrum of tasks dealing with many problems of laminar or turbulent flows in many industrial areas (internal or external aerodynamics, for example) and thank to their versatility they cannot describe some environmental flow pattern (urban areas flow) in demanded details. This is a reason why we have decided to develop a specialized model for the urban area flows. In this model all the approaches that could result in the description of air-flow that will be as exact as possible have been used. For the spatial discretization a finite volume method has been applied together with the Godunov’s approach (Godunov, 1959) used for the advection part of model equations. A MUSCL type of reconstruction (van Leer, 1979) is used for dependent variables together with the TVD (total variation diminishing) temporal integration scheme. For the implementation of the geometrically complicated lower boundary condition so called immersed boundary method (hereinafter abbreviated as ImBM, see Peskin, 1982) has been implemented. At this moment the model is tested as a laminar one without the appropriate turbulent model having been implemented. The vertical temperature stratification is not thought at this moment, too.
The presented contribution is organized as follows. In the first part a description of the used numerical methods is given. The ImBM and its implementation are mentioned in the second part of this contribution. Some preliminary results are described in the third part of this contribution and conclusions together with outlooks dealing with the future developments of the describer basis of the model are contents of the final part.
2. Numerical methods
2.1. Basic set of equations
Using the tensorial indexes the governing equations for unsteady incompressible viscous laminar and isothermal flow can be written in the following form
where i, j = 1,2,3, t is time, xj are the Cartesian coordinates, ui stand for flow velocity vector components, p is a pressure, ρ is a constant density of a fluid (air), Fi denotes other forcing (forces of viscous nature, for example), fi and q are artificial momentum and mass source terms that result from the immersed boundary method utilization (see Kim et al., 2001, for example, more details will by given in the subsequent sections). All dependent and independent variables have been non-dimensioned with a standard way (see Ferziger, Perić, 1997, for example)
, , , , (3)
ã denotes a non-dimensioned variable a, U is a reference value of the flow velocity, D is a reference length scale of the problem under solution. In the following text a tilde symbol above non-dimensioned variable is omitted but all variables are used in the non-dimensional form.
When written in the non-dimensioned form the governing equations are
Re stands for the Reynolds number (Re =), ν denotes the kinematic viscosity coefficient of the fluid (air). As it has been mentioned earlier, at this moment the model deals with the laminar flow thus no turbulence parametrization has been used. The same statement is true for vertical temperature stratification – the impact of it is also not involved into the model equations and the stratification is supposed to be neutral.
2.2. Time integration and spatial discretization
The time-integration method used to solve non-dimensioned eqs. (4) and (5) is based on the fractional step approach (see Brown et al., 2001, for example) where a pseudo-pressure is used to correct the velocity field so that the continuity equation is satisfied at each time step. This method uses the TVD Runge-Kutta 2nd order temporal scheme (see Shu, Osher, 1988) for advection terms of (4) and the Crank-Nicholson implicit scheme for viscous terms. Schematically, this approach can be written in the following form možná bych doplnil indexy a parciální derivace pro konzistenci s předchozím, nebo naopak
, (6) , (7)
where is an advection term and φ is a pseudo-pressure. Superscripts denote time levels when the appropriate terms are evaluated and the symbol u* corresponds to intermediate values of velocity components not-fulfilling the continuity equation constrain.
Spatial discretization uses the staggered finite volume (FV) method with scalar and divergence terms being defined in the centre of FV’s and velocity components on the FV’s sides (see Harlow and Welch, 1965). The whole domain is discretized using the non-equidistant Cartesian grid. In the staggered approach there is not a problem with the pressure gradient terms evaluation like in the collocated one but the most problematic are the non-linear advection terms that can create non-physical wiggles in the areas where high gradients occur. The methods that prevent the creation of these wiggles or those that suppress them are the most advantageous and should be used. In this presented model we have used the Godunov’s type of central scheme proposed by Kurganov and Tadmor, 2000 (abbreviated as KT hereinafter) with the piecewise linear reconstruction of the MUSCL type with the minmod slope limiter to obtain point-wise values u+ and u- on the sides of the adjacent finite volumes. In this kind of the scheme it is not necessary to solve the Riemann problems on the boundaries of finite volumes. This approach is not so accurate as if the classical Godunov’s upwind scheme with Riemann solver has been utilized but it is the more efficient one and is easier to code especially in more dimensional problems. The KT scheme is of the second order of accuracy and do not generate undesirable oscillations and wiggles in the high gradient areas.
For the advection part of the equation (4) that can be written in a form of the hyperbolic law of momentum conservation the following expression can be chosen
A symbol H(u) denotes the numerical flux function evaluated on the boundaries of FV’s that can be written in the form (see KT or Kurganov and Levy, 2000)
In this formula u±(t) denote the values of u evaluated on the boundaries of FV’s from the reconstruction procedure, f (u±(t)) is a flux function (= uiuj) from the eq. (4) and a(t) is a local speed computed from the formula (see KT or Kurganov and Levy, 2000)
where ρ(A) denotes a spectral radius of a matrix A, i.e. ρ(A) = maxi with λi (A) being its eigenvalues. Comparison of several different numerical schemes with that of KT is shown in the following paragraphs.
3. The immersed boundary method
One method how a boundary displaying the complicated geometrical shape can be implemented into a model deals with application of body fitted grid. This approach is frequently used but on the other hand it uses a quite complicated method of the grid generation and, moreover, if the grid is heavily distorted the order of the spatial accuracy of the used numerical method can decrease. A quite effective way how to remedy this problem is brought with the utilization of so called immersed boundary method (ImBM).
The ImBM was firstly used by Peskin, 1982 when modelled the blood flow in a human body vascular system and heart valves. The ImBM enables to use the regular orthogonal structured grid (e.g. Carthesian) even in very complicated shapes with walls moving with the velocity that differs from that of the fluid flow. There exist several methods how to implement the immersed boundary methodology into the models, in the model presented in this contribution the one described with Kim et al., 2001 has been used.
The used ImBM consists in evaluation of the artificial momentum and mass source terms in the equations of motion and continuity. These source terms differ from zero only in those FV’s where the boundary of the obstacle occurs or inside the obstacle, in the space outside of the obstacles (e.g. buildings) the artificial source terms are equal to zero. The principal assumption is that the flow velocity just on the boundary (wall) of the bypassed structure is equal to the velocity of the wall. In case of the buildings, for example, this wall velocity equals zero (no-slip boundary condition). Using some extrapolation procedure it is possible to evaluate the fictitious velocity inside the bypassed body in such a way that on the boundary the velocity is just equal to zero and thus it is possible to evaluate the terms fi in the eqs. (1) and (4). The artificial mass source/sink term q in the continuity equation (2) and (5) is only “active” inside the bypassed body in these volumes close to boundary where only inflow to them or only outflow from them exists (more details in Kim et al., 2001).
4. Some preliminary results
As it has been mentioned in the introductory part of this contribution at this stage the presented results deal with very beginning of the constructed urban air-flow model. From some reasons (less demand on memory, less processor-time consumption) at this stage the 2D approach has been chosen - the transition to the full 3D computation is straightforward and for validation of methods used the 2D approach is sufficient enough.
A standard test problem – so called Sod problem, was used to test the properties of the used KT scheme and comparison of this test results are shown on the subsequent lines. The second step deals with the comparison of the model approach to the analytical (Blasius) solution describing the boundary layer evolution above a flat plate exposed to a homogeneous incoming flow (see Batchelor, 2000). Results of this comparison are presented and discussed in the first paragraph of this chapter. In the next paragraph some results dealing with the lid driven cavity flow are presented and flow around a square cylinder oriented in different ways to it are presented in the last one.
4.1. Sod problem solution
The Sod problem, see, for example, Wesseling, 2001, is a 1D shock tube problem, also called the Riemann problem that is characterized with the initial distributions of pressure, density and velocity. This problem constitutes a particularly interesting and difficult test case, since it presents an exact solution to the full system of 1D Euler equations containing simultaneously a shock wave, a contact discontinuity and an expansion fan. On the Figures 1 – 3 these Sod problem results for the different types of numerical schemes in comparison with the analytical solution are shown. These results deal with solution of the 1D compressible Euler equations in shock tube problem with the initial conditions
The solution is given as a superposition of shock wave and contact discontinuity travelling in the positive x-direction and expansion fan in the negative x-direction. On the Figure 1 values of velocity and density at the model time t = 0,3 for Lax-Friedrichs scheme are shown in the comparison with the analytic solution. In the numerical solution the impact of numerical viscosity smearing the sharp gradient is evident. The second case corresponds to the Lax-Wendroff scheme – Figure 2 – the numerical wiggles are evident in the neighbourhood of the discontinuity both in density and velocity. The result of KT scheme is shown on the Figure 3 where the evident improvement can be seen. On all figures the dashed line represents the analytical solution to the Sod problem and the line with crosses corresponds to numerical solution.
4.2. Boundary layer above a flat plate.
Geometry of this problem is shown on the Figure 4. Bold solid line on the right denotes a flat plate. In some distance from the plate leading edge the boundary layer depth becomes stabilized and does not grow further. For this case the analytical Blasius solution is known and can be compared with the numerical model results. This comparison is showed on the following figures.
Figure 5 shows a comparison of the friction coefficients corresponding to the presented numerical model (+) and Blasius analytical solution (---). The friction coefficient Cf is defined as
where τw is a shear friction on the plate and can be expressed as
μ is a dynamic viscosity coefficient and y is a direction normal to the plate. From the Figure 5 is evident that numerical model results correspond to the analytical ones far distant from the plate leading edge, close to that point there are some discrepancies but they exists in the non-stabilized region.
Figure 6 show profiles of (a) u- and (b) v-components of velocity at the distance x = 0,5 from the plate leading edge. In the y direction the non dimensional coordinate η has been used
y is a dimensional coordinate normal to the plate, U is incoming velocity, x is a coordinate along the plate and ν is a kinematic viscosity coefficient. From the Figure 6 a) a good correspondence between numerical and analytical solutions can be seen in the case of u-component of velocity. Some differences are seen on the Figure 6 b) showing the v-velocity components normal to the plate. A reason of them could originate in the y direction boundary condition that has been applied in the numerical approach.
4.3. Upper lid driven cavity flow
This problem deals with modelling of flow in cavity that is driven with the moving upper lid. This is the example of the simplest urban canyon circulation. There are several well described results of this situation – see, for example, Ghia et al., 1982, Erturk et al., 2005. The schematic geometry of this case together with the boundary conditions is shown on the following Figure 7.
The cavity circulation structure depends on Reynolds number Re = of the problem where U is the upper lid non-dimensional velocity and D is the cavity size (in the case of the square cavity it is the length of the cavity side). The structure of circulation consists from primary vortex and, depending on the value of Re, from the secondary and tertiary vortices (when the Re is high enough). The circulation patterns are shown on the Figures 8 (Re = 100) and 9 (Re = 5000). The velocity on the upper boundary is directed from the left. In the both lower corner the secondary vortices can be seen in this case with Re being equal to 100. When the value of Re increases the axis of the primary vortex moves close to the centre of the square and in both lower corners secondary vortices intensify and tertiary vortices appear. Another secondary vortex appears on the upper right corner. The circulation structure for Re = 5000 is shown on the Figure 9.
On the next figures (Figures 10 - 13) comparisons of the presented model results with those published in Ghia et al., 1982 or Erturk et al., 2005 are presented. For the Re equal to 100 (Figures 10 and 11) both velocity components (u and v) correspond to those of Ghia et al., 1982 almost exactly. The velocity profiles are shown for the location corresponding to the mid of square sides (x = 0,5 for u velocity component and y = 0,5 for v velocity component). The similar figures but for Re = 5000 are indicated as Figures 12 and 13. For this situation some differences can be seen. A possible reason for these differences could come from, firstly, the different numerical methods used and, secondly, the different resolution used. But, despite those differences, from all the figures is evident, that the circulation in both cases are almost equivalent with compared results of Ghia et al., 1982 and Erturk et al., 2005.
4.4. Flow around a square
A square can be taken as the first approximation to the building. Modelling of a flow around a square that has been oriented with a different way for several values of Re has been performed. Geometry of the problem is schematically shown on the following Figure 14. A letter H denotes the width of the area, d is the projection of the square on the plane perpendicular to the incoming flow and α is an angle of the square orientation with respect to the incoming flow. Before we start to describe results some parameters used for it will be mentioned.
An important parameter that can also affect results both in mathematical and physical modelling is a ratio β (see Sohankar et al., 1995) defined as
β = . (17)
Another important criterion dealing with the speed of vortices generation is the Strouhal number St (Strouhal, 1878)
f is a shedding frequency. Forcing acting by the body (square) on the fluid flow can be described with two parameters. The first one is a drag coefficient CD
where FD denotes the resistance force of the square to the fluid flow. The second one is a lift coefficient CL
FL is the force acting on the square in the direction perpendicular to the incoming flow.
Character of the flow around the square depends on the value of Re. In very low values (Re < 1) streamlines simply embrace the square. When the value of Re increases the recirculation zones appear and when Re are larger than 50 – 55 the zones become unstable and individual vortices separate in the wake of the square. A dependence of the recirculation zone length on the Re is shown on the Figure 15. The results are shown for described model ones and two values of ratio β and compared with those presented in Breuer et al., 2000 where this ratio is equal to . For Re less than 250 the flow can be considered as 2D but some disturbances infringing on this 2D character appear around Re equal to 150 – 175 (see Saha et al., 2003). When the value of Re is greater than 300 the flow becomes turbulent.
On the following figures some comparisons of this model results with those presented by other authors are shown. The Figure 16 shows the dependence of the Strouhal number on the Reynolds number and on the Figure 17 a dependence of the mean value of CD on the Reynolds number is depicted. Both figures are valid for orientation of the square with α = 0°. A temporal development of CL is shown on the Figure 18 for Re = 100 and α = 0°.
Flow patterns for Reynolds numbers equal to 30 and 200 and square orientation characterized with α being equal to 0° and 45° are shown on the following Figures 19 – 22. Lines are flow field streamlines. In a lower Re value a stationary recirculation zones exist in the wake of the square, when Re equals 200 the vonKármán vortex street generation in the wake of square can be seen.
5. Concluding remarks and future outlook
In this contribution some preliminary results of a new model of fluid flow in complex geometry (urban) areas are shown. At this moment results deal with the 2D approach but transition to the full 3D approach is straightforward. The 2D approach was chosen mainly because at each time step it is necessary to solve elliptic partial differential equation for pressure corrections ant this is a very time consuming procedure. It will be necessary to use some more efficient elliptic equation solver for solving the full 3D system. But also in the 2D approach it was possible to compare the obtained results with those published earlier. There are some differences and it will be necessary to find out the reasonable explanation of them. But, on the other side, obtained results seem very promising and we will continue in creation of the presented model using the mentioned high resolution methods. To demonstrate the efficiency of the high resolution used for advection part of the equation the comparison of the numerical methods results with the analytical ones of the Sod problem has been shown. From this comparison can be seen that Kurganov Tadmor scheme used exhibits neither numerical wiggles in the high gradient areas nor smearing these gradients.
In the close future the two steps are necessary to perform: The first one is to find out some more efficient pressure correction solver and to go over to the full 3D system. The second one deals wit the parametrization of turbulence. It will be necessary to apply some efficient turbulence model for the case characterized with the higher values of the Reynolds number. And then also the vertical thermal stratification will be necessary to implement into the equations as now the fluid is supposed to be neutrally stratified.
This research has been supported by the Grant Agency of the Czech Academy of Sciences, grant no. T400300414 and by the Grant Agency of the Czech Republic, grant no. 205/06/0727.
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Figure 1 Sod problem. The analytical solution (dashed line) and the Lax-Friedrichs scheme solution (line with crosses) comparison.
Figure 2 The same as on the Figure 1 but for the Lax-Wendroff scheme (line with crosses).
Figure 3 The same as on the Figure 1 but for the Kurganov Tadmor scheme (line with crosses).
Figure 4 Boundary layer evolution above a flat plate.
Figure 5 A comparison of friction coefficients; + numerical model, --- Blasius analytical results.
Figure 6 Profiles of velocity components; --- Blasius results, + numerical model results; a) u-component, b) v-component of velocity.
Figure 7 Schematic geometry and boundary conditions of the upper lid driven cavity flow.
Figure 8 Upper lid driven cavity flow. White lines – streamlines, colour corresponds to the absolute value of the non-dimensional flow velocity, Re = 100.
Figure 9 Upper lid driven cavity flow. White lines – streamlines, colour corresponds to the absolute value of the non-dimensional flow velocity, Re = 5000.
Figure 10 A profile of the u-component of velocity at x = 0,5 for Re = 100. Results of Ghia et al., 1982 denoted by +.
Figure 11 A profile of the v-component of velocity at y = 0,5 for Re = 100. Results of Ghia et al, 1982 denoted by +.
Figure 12 A profile of the u-component of velocity at x = 0,5 for Re = 5000. Results of Ghia et al., 1982 denoted by +, those of Erturk et al., 2005 denoted by ×.
Figure 13 A profile of the v-component of velocity at y = 0,5 for Re = 5000. Results of Ghia et al., 1982 denoted by +, those of Erturk et al., 2005 denoted by ×.
Figure 14 Geometry of the flow around the square.
Figure 15 Dependence of the recirculation zone length on Re with different values of β and α = 0°.
Figure 16 Dependence of St on Re for square orientation with α = 0°.
Figure 17 Dependence of CD on Re for square orientation with α = 0°.
Figure 18 A temporal development of CL for square orientation with α = 0°.
Figure 19 Flow around square for Re = 30, α = 0°, schematically shown instantaneous streamlines.
Figure 20 Flow around square for Re = 30, α = 45°, schematically shown instantaneous streamlines.
Figure 21 Flow around square for Re = 200, α = 0°, schematically shown instantaneous streamlines.
Figure 22 Flow around square for Re = 200, α = 45°, schematically shown instantaneous streamlines.
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