Lid-Driven Cavity 2D, Re = 10000, 600 x 600 Cells: Kinetic Energy and Vortex Volume Fraction.

[@Srinivasa-no-eqn-without-god]

THANKS SIR

[@CFD-Bielefeld]

Due to numerous requests, I will give a series of technical tips on calculating the vortex volume fraction in this comment section.

[@CFD-Bielefeld]

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 lid is equal to zero. Therefore, the Reynolds-Number Re and the kinetic energy are initially zero.

[@CFD-Bielefeld]

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. For t in [0,1], the lid velocity is acceleratated from u = 0 to u=100. For t > 1, u is constantly equal to 100 and as a result Re is constantly equal to 10.000= 1e+05.

[@CFD-Bielefeld]

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 Frobeniusnorm. 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.

[@Srinivasa-no-eqn-without-god]

Excellent sir for your valuable contribution, sir what is the purpose of vortex volume fraction, can we find for laminar flow say Re

[@CFD-Bielefeld]

The vortex volume fraction VVF is a dimensionless global parameter and a measure of how strongly the flow is dominated by vortices. The VVF can be calculated for laminar as well as for turbulent flows. In the case of a rigidly rotating fluid with constant angular velocity, which is a purely laminar flow, the VVF will be equal to 1 and this means that the vortex covers the entire flow area. In contrast, in a pure shear flow VVF is equal to 0. In the Lid-Driven Cavity, the VVF depends on the Reynoldsnumber. Numerical experiments for 1000 < Re < 10000 show, that VVF increases with Re, what is intuitively expected. For my definition of VVF, see the video "Lid-Driven Cavity: Vortex Volume Fraction", September 08, 2024. Thank you for watching.

[@Srinivasa-no-eqn-without-god]

But sir what is Re and why with less time it is steady state