First things first: We start the time-dependent simulation at time t = 0 with an initial state, where the velocity field is equal to zero in the whole fluid domain. Even the velocity of the 4 sidewalls is equal to zero. Therefore, the Reynolds-Number Re and the kinetic energy are initially zero.
The Reynolds Number is defined as follows: Re = u*h*rho/eta. Therein u is the lid-velocity, h is the height of the cavity, rho is the density and eta is the dynamic viscosity of the fluid. Since the lid-velocity is a function of time, the Reynolds Number is a function of time too. All sidewalls have the same velocity magnitude. For t in [0,1], the lid velocity is acceleratated from u = 0 to u = 1. For t > 1, u is constantly equal to 1 and as a result Re is constantly equal to 100.
How to calculate the Volume Vortex Fraction: First, the velocity gradient L := nabla v, which is a rank 2 tensorfield, is calculated in each node of the grid. The cartesian coordinates of L are: L_ij := d_i v_j ( i = 1,2,3 ; j = 1,2,3 ) where d_i is the partial derivative operator in the i-direction and v_j is the j-coordinate of the velocity vector. In the next step we decompose the tensor L into a symmetric part S and an antisymmetric part W: S := 1/2 ( L + L^T ) and W := 1/2 ( L + L^T ) S_ij := 1/2 ( L_ij + L_ji ) and W_ij := 1/2 ( L_ij - L_ji ) In the next step we calculate the quantity Q in each node as follows: Q := 1/2 ( ||W||^2 - ||S||^2 ) where ||.|| is the Frobebniusnorm. To calculate ||S||^2, you simply have to square all elements of S and add them up. Proof that: ||W||^2 = 1/2 | curl v |^2 Next, we normalize the scalar field Q: Qn := 1 if Q > 0 Qn := 0 if Q 0 holds.
Due to numerous requests, I will give a series of technical tips on calculating the vortex volume fraction in this comment section.
First things first:
We start the time-dependent simulation at time t = 0 with an initial state, where the velocity field is equal to zero in
the whole fluid domain. Even the velocity of the 4 sidewalls is equal to zero.
Therefore, the Reynolds-Number Re and the kinetic energy are initially zero.
The Reynolds Number is defined as follows:
Re = u*h*rho/eta.
Therein u is the lid-velocity, h is the height of the cavity,
rho is the density and eta is the dynamic viscosity of the fluid.
Since the lid-velocity is a function of time, the Reynolds Number
is a function of time too.
All sidewalls have the same velocity magnitude.
For t in [0,1], the lid velocity is acceleratated from u = 0 to u = 1.
For t > 1, u is constantly equal to 1 and as a result Re is constantly
equal to 100.
How to calculate the Volume Vortex Fraction:
First, the velocity gradient L := nabla v, which is a rank 2 tensorfield,
is calculated in each node of the grid.
The cartesian coordinates of L are:
L_ij := d_i v_j ( i = 1,2,3 ; j = 1,2,3 )
where d_i is the partial derivative operator in the i-direction and v_j is the j-coordinate of the velocity vector.
In the next step we decompose the tensor L into a symmetric part S and an antisymmetric part W:
S := 1/2 ( L + L^T ) and W := 1/2 ( L + L^T )
S_ij := 1/2 ( L_ij + L_ji ) and W_ij := 1/2 ( L_ij - L_ji )
In the next step we calculate the quantity Q in each node as follows:
Q := 1/2 ( ||W||^2 - ||S||^2 )
where ||.|| is the Frobebniusnorm. To calculate ||S||^2, you simply have to square all elements of S and add them up.
Proof that:
||W||^2 = 1/2 | curl v |^2
Next, we normalize the scalar field Q:
Qn := 1 if Q > 0
Qn := 0 if Q 0 holds.