Two phase flow
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The starting point in Lagrangien approach is the fundamental law of dynamics: | The starting point in Lagrangien approach is the fundamental law of dynamics: | ||
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| - | + | mp frac{dv}{\dt} = - U(t) \frac{dt}{\tau} + \frac{2 u'}{\tau}^{1/2} dW(t) | |
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Revision as of 11:07, 24 September 2007
Contents |
Introduction
article in progress
Importance of two phase flow in industrial configurations
Two phase flow phenomena occur in various industrial application in all fluid mechanics application fiels. Aerospace, automotive, nuclear applications, etc. In all this domain, prediction of two phase behaviour is important. Prediction of liquid spray in an internal combustion engine should enable us to have a better control on combustion process and then to reduce pollutant emissions. Controlling water - steam equilibrium in a coller system enable to prevent from industrial accident, de-icing/anti-icing of aircraft on the ground etc. Any other examples can be quoted here.
Overview of the different available approach
Two main family can be distinguished to model two phase flow, depending of the two phase configuration approach. In case of dispersed configuration a lagrangian approach is suitable. Such an approach consists in following dropplets (or bubbles) during then movement. This is done by applying external force on the particle and solving acceleration, then velocity and finally position. On the other hand, two phase flow can be solve with an eulerien approach. As in all eulerian framework, this approach consists in considering inlet and outlet flux in a given volume. In such an eulerian approach, two family can be distinguished : Mixture model and Two fluids model, those two approach will be detailed in corresponding section bellow.
Lagrangian dispersed two-phase flow modelling
The main goal in Lagrangien approaches is to statiscally particle history in given flow fields. The starting point in Lagrangien approach is the fundamental law of dynamics:
- Failed to parse (unknown function\dt): mp frac{dv}{\dt} = - U(t) \frac{dt}{\tau} + \frac{2 u'}{\tau}^{1/2} dW(t)
dispersed phase was treated using the lagrangian reference frame; the main goal of this approach is to statistically represent particle history in given flow fields. The trajectories of the particles are computed on the basis of the equation of motion corrected by Odar and Hamilton [12]. This equation was established with the following assumption:
-The particles are rigid sphere.
-The particles are not rotating.
-Particle-particle interactions are neglected.
It was demonstrated that the dominant forces are the drag and the gravity. The equation of motion can be written as:
The mathematical expression for the drag coefficient is:
The instantaneous fluid velocity u is decomposed into a mean part which is known (from turbulence model prediction) and a fluctuating part u’. In 2D description, we have to determine a vector of correlated random variables: U’= (ui'(0), uj’ (0)... ui’ (nt), uj’ (nt) ...) where u’(nt) is the fluid fluctuating velocity at time nt. U’ is determinate Starting from a vector Y(yi) of non correlated random numbers with a Gaussian distribution (<yi>=0 and<yiyj>=ij), a linear relation is assumed between U’ and Y, that means a matrix B* is defined such as: U’=B*Y. Recalling that B* is symmetric and positive definite, a Cholesky factorization leads to: R=B* TB* where TB* is the transposed B* matrix and R is a correlation matrix. We thus get the (b*ij) elements and then the vectors U’. Lagrangian correlation coefficients are expressed by Frenkiel function [6]:
where m is a loop parameter and τL is the Lagrangian integral time scale. The loop parameter m should be considered as a closure constant to be determined with the aid of experiments and the recommended value is m=1(Picart et al. [13]). The Fluid particle trajectory and the discrete particle trajectory are different so we have to release simultaneous a fluid trajectory and a particle trajectory. Figure 2 presents an overview of the method.
Figure 2: Coupled fluid and particle trajectories on one time step. Initially on the same location, the fluid particle F is moved to location xf(t+t) using one step stochastic procedure with respect to the fluid Lagrangian integral time scale. The discrete particle trajectory P is calculated with its motion equation. The fluid velocity is then transferred from the fluid position to the particle position to the particle position xp(t+t) with respect to Eulerian correlation. The process is then repeated. That approach has been extended by Berlemont et al [3] and includes the correlation matrix method for fluid trajectories in order to handle any kind of correlation. The method is described on Figure 3. A fluid particle is simultaneously followed with the discrete particle on several time steps. The fluid velocity is transferred from the fluid particle location to the discrete particle location by use of Eulerian correlations.
Figure 3: Coupled fluid and particle trajectories and correlation domain.
This process is carried out until the discrete particle leaves a correlation domain defined around the fluid particle. When the distance r between the two particles is greater than the correlation length scale LD, a new fluid particle is sampled on the discrete particle location and the process is repeated.
This scheme is imaging the crossing trajectory effects in a very physical way.
The Eulerian correlation between fluid particle and discrete particle can be written as:
Where is the Eulerian length scale formed by:
The correlation length scale LD is given by:
It is important to keep in mind that particle dispersion is roughly proportional to the turbulence time/length scales. The instantaneous particle location is obtained from the definition of the velocity:
