Diagnóstico - Control de conformidad
ESTADO 2 frenado dinámico (por encima de
In this section, the numerical procedure for solving Navier-Stokes equations based on FFD will be discussed. The proposed solver in the present work, named CityFFD, is based on the dimensionless equations. The governing equations, mass and momentum equations, in the dimensionless forms are: ∇ · 𝑈 = 0 (2-1) 𝜕𝑈 𝜕𝑡 + (𝑈 · ∇)𝑈 = −∇𝑝 + 1 𝑅𝑒∇2𝑈 + 𝑓 (2-2)
The conventional pressure-based semi-Lagrangian method comprises four main steps: 𝜕𝑈 𝜕𝑡 = 1 𝑅𝑒∇2𝑈 + 𝑓 (2-3) 𝜕𝑈 𝜕𝑡 + (𝑈 · ∇)𝑈 = 0 (2-4) ∇2𝑝 = 1 ∆𝑡𝛻. 𝑈 (2-5) 𝜕𝑈 𝜕𝑡 = −𝛻𝑝 (2-6)
22
1. Navier-Stokes equation, Eq. (2-2), is first reduced to diffusion and source terms, in the absence of pressure and advection terms, and is solved to calculate an intermediate velocity field for the diffusion and source terms only (Eq. 2-3).
2. Equation of the advection term (Eq. 2-4) is then solved by the semi-Lagrangian method to obtain the 2nd intermediate flow filed.
3. Poisson equation of pressure (i.e. pressure correction) is then solved for correcting the pressure field (Eq. 2-5).
4. Based on the pressure corrections, the velocity field is corrected to satisfy the condition of the divergence-free flows (Eq. 2-6).
In the above procedure, both the equations of the diffusion term (Eq. 2-3) and the pressure term (Eq. 2-5) are discretized based on the central 2nd-order of accuracy. These equations are elliptic, so an iterative solver is often needed. Here a three-level V-cycle multigrid method is used to speed up the calculation of the diffusion and Poisson equations (Appendix 1).
To explain the process of solving the Eulerian advection equation, Eq. (2-4), from the Lagrangia n perspective by the semi-Lagrangian method, wIe start with the advection equation of a general scalar term, ∅, along its characteristic curve:
𝑑∅ 𝑑𝑆 = 0
(2-7) where ∅ = 𝑈 for the linear momentum conservation equation and 𝑆 can be written as follows:
𝑑𝑆 = 𝑈𝑑𝑡 → 𝑆𝑛≈ 𝑆𝑛+1 − 𝑈∆𝑡 (2-8)
Here, 𝑛 + 1 and 𝑛 are the new and current time steps, respectively. Using a first-order temporal discretization, Eq. (2-7) becomes:
23 ∅𝑆𝑛+1|𝑋
𝑎 ≈ ∅𝑆𝑛|𝑋𝑑 (2-9)
Because ∅𝑆𝑛 +1|𝑋
𝑎 is always for the time 𝑛 + 1 and ∅𝑆𝑛|𝑋𝑑 for 𝑛 , for simplification and generalization, the time terms is dropped so Eq. (2-9) becomes:
∅𝑎 ≈ ∅𝑑 (2-10)
Eq. (2-10) shows that ∅ is considered the same at both arrival and departure points within one time step. To calculate ∅𝑑, an interpolation scheme based on the neighbor Eulerian cells is needed (see Figure 2-2).
Figure 2-2 Procedure of the semi-Lagrangian method with linear interpolation scheme.
Conventional FFD is based on a linear interpolation scheme which is low-order of accuracy. First, the departure point position is found. To find the position, we move backward to the location, 𝑋𝑑, at a distance of one time step times the current flow velocity, 𝑈𝑎∆𝑡, from the arrival point location, 𝑋𝑎, along the characteristic curve:
𝑋𝑑 = 𝑋𝑎− 𝑈𝑎∆𝑡 (2-11)
In the next step, the neighbor cells are found. In Figure 2-2, 𝑁𝑗 (the Eulerian neighbor cells) shows the neighbor cells around the departure point. Using an interpolation scheme, the fluid property
24
values at the departure point are calculated. Eq. (2-12) shows the linear interpolation scheme (2nd- order of accuracy) which is used to find the value of ∅𝑑 at the departure point:
∅𝑎 = ∅𝑑 = ∑ 𝑤𝑗∅𝑁 𝑗 4
𝑗=1
(2-12)
where the 𝑤𝑗 values are the weights of the interpolation scheme, e.g. for 2-D setup, bilinear interpolation can be applied.
Low-order interpolation schemes thus often cause inaccurate predictions of fluid properties, so, poor conservation of advected properties and/or high numerical dissipations (Zerroukat, 2010; Zuo et al., 2012). Another thing which affects the accuracy is the position of the departure point and its distance from its neighborhood Eulerian cells. My study in the next chapters shows that the magnitude of truncation error is smaller if the departure point is closer to the Eulerian grids.
On the other hand, conventional FFD is based on constant velocity assumption for constructing the characteristic curves. This assumption may provide accurate results by using small time step, but the huge numerical error, especially for the transient problems, can be seen for the large time steps. Using a large time step creates considerable deviation between the location of the estimated departure point and the accurate position of the point (see Figure 5-1).
The conventional semi-Lagrangian solver is pressure-based (PB) and by solving the Poisson equation, pressure domain is calculated and free divergence condition is satisfied. Pressure-based solver originally developed by Harlow and Welch (1965) for unsteady flow problems and then later extended by Patankar and Spalding (1972) for steady-state calculations. In this method, computations of the velocity and the pressure fields are decoupled (Blazek, 2005). Here, the pressure domain is calculated by solving the Poisson equation and then the velocity is obtained by
25
the pressure domains. On irregular grids, solving the Poisson equation is not straightforward (Min and Gibou, 2006). In addition, solving the Poisson equation often needs iterative solvers, making it a real challenge for parallelizing FFD-based solvers on modern high-performance computing infrastructures, such as OpenMP and/or GPU, especially for large data cases (e.g. tens of millio ns of grids), which may only be solved in a parallel manner.
In this thesis, four novel ideas are proposed to overcome the drawbacks of conventional FFD models and a new FFD model, so-called CityFFD, is developed for modeling urban microclimates based on the coarse grids and large time steps. Chapter 3 is for solving the coarse grid problem by proposing a new high-order interpolation scheme which is capable to highly control the numerica l errors. Chapter 4 is devoted to the time step size and its impact on the accuracy. Here it is shown different time steps can significantly affect the magnitude of truncation errors. In chapter 5, a new model is proposed to improve the accuracy of FFD by using a high-order temporal method. The proposed method can provide accurate results even by using large time steps. Chapter 6 investigates the computational cost of the different components of FFD. In this section, I show by removing the most time-consuming part of the conventional FFD method, the Poisson equation, it is possible to speed up the simulation at least three times. The proposed method is also better for parallelization techniques. In chapter 7, LES model is added to CityFFD for modeling the turbulence of microclimate problems. In the end, chapter 8 is dedicated for an application case. Here, a real city is simulated and the importance of urban microclimate on the indoor environme nt and building energy/thermal performance is demonstrated. It is achieved by integrating CityFFD with another newly developed model, CityBEM, which is a fast building energy model.
26