Changes between Version 1 and Version 2 of doc/app/examples/dns

Timestamp:

Jun 18, 2019 2:02:23 PM (6 years ago)

Author:

Giersch

Comment:

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doc/app/examples/dns

@@ -1 @@
-An example parameter set for DNS runs will follow. 
+This chapter provides an example for a DNS validation run, where free convection over a heated plate was investigated. The reference for this simulation is case NsD from [#mellado2012 Mellado (2012)]. To save computational costs, the computationally cheaper case described in the appendix of [#mellado2012 Mellado (2012)] is attached. In the following, some background knowledge is given, to explain the settings in the attached _p3d file. 
+
+Apart from RANS and LES simulations, PALM can also be used to perform Direct Numerical Simulations (DNSs). In contrast to the RANS mode, there is no need to adjust PALM's system of equations. DNSs can be performed by just using appropriate settings in the _p3d file. The following terms in the equation of motion differ from each other depending on the used mode (left DNS, right LES mode, for more info see also [wiki:doc/tec/gov#Governingequations governing equations] and [wiki:doc/tec/sgs#Deardorffsubgrid-scalemodel subgrid-scale model]):
+
+{{{
+#!Latex
+\begin{align*}
+ - \frac{1}{\rho_0} \frac{\partial p^\ast}{\partial x_i} &\neq - \frac{1}{\rho_0} \frac{\partial \pi^\ast}{\partial x_i} = - \frac{1}{\rho_0} \frac{\partial}{\partial x_i} (p^* + \frac{2}{3} \rho_0 e), \\
+\nu_m \frac{\partial^2 u_i}{\partial x_k^2} &\neq -\frac{\partial}{\partial x_j} \left(\overline{u_i^{\prime\prime} u_j^{\prime\prime}} - \frac{2}{3}e\delta_{ij}\right) \underbrace{=}_{SGS-model}  -\frac{\partial}{\partial x_j} -K_\mathrm{m} \left(\frac{\partial u_{i}} {\partial x_{j}} + \frac{\partial u_j} {\partial x_{i}}\right) = K_\mathrm{m} \left( \frac{\partial^2 u_{i}}{\partial x_j^2} +  \frac{\partial}{\partial x_i} \underbrace{\frac{\partial u_{j}}{\partial x_j}}_{=0} \right) = K_\mathrm{m} \frac{\partial^2 u_{i}}{\partial x_k^2}, 
+\end{align*}
+}}}
+
+with ''π^∗^ = p^∗^ + 2/3 ρ,,0,, e '' (the overbar indicating filtered quantities in LES mode is omitted ). Similar, also each convection–diffusion equation differs in one term (example for the potential temperature is shown):
+
+
+{{{
+#!Latex
+\begin{align*}
+\nu_h \frac{\partial^2 \theta}{\partial x_j^2} \neq - \frac{\partial}{\partial x_j} \overline{u_j^{\prime\prime} \theta^{\prime\prime}} \underbrace{=}_{SGS-model} K_\mathrm{h} \frac{\partial^2 \theta}{\partial x_j^2}. 
+\end{align*}
+}}}
+
+Therefore, the governing equations implemented in the code are valid for DNS if a constant [../../inipar#km_constant eddy diffusivity] (''K'',,m,,) is used for the flow. In this way, no TKE is computed (''e''=0) and the not equal signs from the equation above change to equal signs. For DNS, the term eddy diffusivity is not valid anymore. Instead, we specify with [../../inipar#km_constant km_constant] the kinematic viscosity ''ν'',,m,, and molecular diffusivity ''ν'',,h,, for heat. In the medium air, these values are roughly 1.76e-5 if a [../../inipar#prandtl_number Prandtl number] of 1 is used. The grid spacings Δ need to be comparable to the Kolmogorov length characterizing the smallest scales inside the turbulent region to ensure a DNS. With the general assumption that the Kolmogorov length is comparable to the diffusion length ''z'',,0,, characterizing the diffusive sublayer next to the wall, we can estimate Δ from, e.g. (see [#mellado2012 Mellado (2012)])
+
+{{{
+#!Latex
+\begin{align*}
+\Delta = z_0 = \left( \frac{\nu_h^2}{b_0} \right)^{1/3}
+\end{align*}
+}}}   
+
+with ''b'',,0,, being the surface buoyancy, which is given from the boundary condition. Due to the fine grid spacing, only several tens of seconds can be simulated. A simulation with an [../../inipar#end_time end time] of 10s, 1024x1024x768 grid points, and grid spacings of 0.86e-3m needs about 1200 core hours on 512 CPUs already. Due to this short simulated time, the Coriolis force is switched off. 
+
+=== References ===
+
+* '''[=#mellado2012 Mellado, J. P.], 2012:''' Direct numerical simulation of free convection over a heated plate. ''J. Fluid Mech.'', '''712''', 418-450.