Category Archives: CFD

Drop evaporation and phase change

the equations we solve are two: Mass conservation for each chemical species i and phase k :

\frac{\partial\rho_{k}\Upsilon_{ki}}{\partial t} + \nabla\cdot \left(\rho_{k}\Upsilon_{ki}\vec{u_{k}} - \rho_{k}D_{ki}\nabla\Upsilon_{ki}\right)=0

and energy conservation for each phase:

\frac{\partial\rho_{k} E_{k}}{\partial t} + \nabla\cdot\left( \rho_{k} E_{k}\vec{u_{k}}+p_{k}\vec{u_{k}}+\vec{q_{k}}\right)= \bar{\tau}:\nabla\vec{u_{k}}

 

where \Upsilon_{ki} is the mass fraction of phase k of species i and the others are the standard notation for density, pressure,velocity,time,energy,heat flux and diffusion coefficient. Moreover the flow is considered isobaric and the viscous dissipation is neglected.

The algorithm that solves the problem is included in the document.

phase-change-submodel

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Lax-Wendroff scheme

The shock wave propagation equation and numerical approach. The governing equation is:

\frac{\partial \mathbf{q}}{\partial x}+\frac{\partial \mathbf{F}}{\partial x} = 0

with: \mathbf{q}=\left[ \rho,\rho u, p/\gamma(\gamma-1) + \frac{\rho}{2}u^2 \right]^T  and  \mathbf{F}=\left[ \rho u,\rho u^2 +p/\gamma, [p/(\gamma-1)+ 0.5\rho u^2]u \right]^T

where \rho,p,u are the (non-dimensional) density, pressure and velocity.

FINITE VOLUME METHODS

   The functions of interest (the ones expanded with Taylor formula) give a big remainder that is not smaller than the first terms. Moreover it is necessary to conserve the integral of such functions along the computational domain. These demands lead to a design of FV methods that are basically arise from Finite Difference methods. The physical domain is considered to be divided into cells. Between time t^n and time t^{n+1}, the increment of a physical (extensive) quantity, let us say the mass in cell C_{j}, denoted by:

mass_{j} = meas(C_{j}) density_{j}

is given by the sum of flux $latex flux_{jk}$ exchanged with each neighbouring cell C_{j} :

mass_{j} = mass_{j} + \sum_{k\in N(j)} flux_{jk}  (1)

Conservation of total mass is ensured by the equality:

flux_{jk} = - flux_{kj} (2)

In the case of linear conservation laws, this may be written, for example, as:

meas(C_{j}) u^{n+1} = meas(C_{j}) u^{n+1} + \sum_{k\in N(j)}(t^{n+1} - t^{n}) Hv(u_{j}, u_{k})

in which the elementary flux is assumed to satisfy a consistency equality and to approximate the normal flux of uV through the common interface of the two cells:

Hv(u_{j}, u{j}) = \int_{\partial C_{j}\cap \partial C_{j}} Vnu_{j} d \sigma

The Lax-Wendroff theorem for steady linear case can be summarized as:

If the scheme is conservtive according to the flux definition (1),(2), and if the discrete solution converges to a limit, then the limit is a weak solution ( i.e a distribution satisfying the equations).

Accuaracy is influenced by the specific method of discretization and cell type. In cell-center schemes primal cells give the partition. In vertex-centered schemes a dual partition is used for flux-derivation.

Lax-Wendroff scheme was intruduced as the first stable three-point scheme applicable for compressible flow problems. It is commonly used at hyperbolic partial differential equations. It is second-order accurate in both space and time. This method is an example of explicit time integration where the function that defines governing equation is evaluated at the current time. The first step in the Lax–Wendroff method calculates values for the quantity at half time steps, t_{n+1/2} and half grid points, x_{j+1/2}. In the second step values at t_{n+1} are calculated using the data for t_{n} and t_{n+1/2}.

1) LAX-WENDROFF scheme:

q^{{}*{}}_{j+1/2} = 0.5(q^{n}_j + q^{n}_{j+1}) - 0.5\frac{\Delta t}{\Delta x}[F^{n}_{j+1}-F^{n}_{j}]

q^{n+1}_j = q^{n}_j - 0.5\frac{\Delta t}{\Delta x}[F^{{}*{}}_{j+1/2}-F^{{}*{}}_{j-1/2}]

2) MACCORMACK scheme:

q^{*}_j = q^{n}_j - \frac{\Delta t}{\Delta x}(F^{n}_{j+1} -F^{n}_{j})

q^{n+1}_j = 0.5(q^{n}_j + q^{{}*{}}_j) - 0.5\frac{\Delta t}{\Delta x}[F^{n}_{j}-F^{n}_{j-1}]

Shock Wave propagation with pressure ratio = 2.5 and γ =1.4

Shock Wave propagation with pressure ratio = 2.5 and γ =1.4

 

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Why isn’t open source CFD solution for everyone?

a nice article here  about open source CFD. I agree with most of it. I especially stand for the following:

The solution may be NOT  for you,  if:

  • you are doing commercial projects, but you feel CFD license is too expensive;

  • nobody in the team has direct (not just self-claimed) experience with programming experience in C/C++/FORTRAN;

  • you are imprisoned  in a Windows-only environment (maybe by your IT dept.)

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Super Highway Convection

The following experiment  about  the thermally stressed layer of a binary liquid mixture. Recently, researchers were exploring this problem—with the added twist of tilting the fluids a few milliradians when they discovered a surprising result. After an extended time, the convection self-organized into alternating parallel columns of ascending (dark) and descending (light) fluid. The researchers nicknamed this behavior super-highway convection.

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