Intro

The classical de Vahl Davis benchmark test is defined for the natural convection of the air ($\Pr =0.71$) in the square closed cavity (${{\text{A}}_{\text{R}}}=1$). The only physical free parameter of the test remains the thermal Rayleigh number. In the original paper [1] de Vahl Davis tested the problem up to the Rayleigh number ${{10}^{6}}$, however in the latter publications, the results of more intense simulations were presented with the Rayleigh number up to ${{10}^{8}}$. Lage and Bejan [2] showed that the laminar domain of the closed cavity natural convection problem is roughly below $\text{Gr1}{{\text{0}}^{9}}$. It was reported [3, 4] that the natural convection becomes unsteady for $\text{Ra}=2\cdot {{10}^5}$. Here we present a MLSM solution of the case.

\begin{equation} \text{Ra}\text{=}\,\frac{\left| \mathbf{g} \right|{{\beta }_{T}}\left( {{T}_{H}}-{{T}_{C}} \right){{\Omega }_{H}}^{3}{{\rho }^{2}}{{c}_{p}}}{\lambda \mu } \end{equation} \begin{equation} \text{Pr}=\frac{\mu {{c}_{p}}}{\lambda } \end{equation}

[1] de Vahl Davis G. Natural convection of air in a square cavity: a bench mark numerical solution. Int J Numer Meth Fl. 1983;3:249-64.

[2] Lage JL, Bejan A. The Ra-Pr domain of laminar natural convection in an enclosure heated from the side. Numer Heat Transfer. 1991;A19:21-41.

[3] Janssen RJA, Henkes RAWM. Accuracy of finite-volume disretizations for the bifurcating natural-convection flow in a square cavity. Numer Heat Transfer. 1993;B24:191-207.

[4] Nobile E. Simulation of time-dependent flow in cavities with the additive-correction multigrid method, part II: Apllications. Numer Heat Transfer. 1996;B30:341-50.

. Figure 1: Scheme of the de Vahl Davis benchmark test

Code

Full examples can be found under the examples in the code repository.

Explicit ACM method with CBS looks

    v2[boundary] = vec_t{0.0, 0.0};
    T2[left] = O.T_cold;
    T2[right] = O.T_hot;
    //Time stepping
    for (int step = 0; step <= O.t_steps; ++step) {
        for (int i_count = 1; i_count < _MAX_ITER_; ++i_count) {
            // Navier Stokes
            for (auto c : interior) {
                v2[c] = v1[c] + O.dt * (-1 / O.rho * op.grad(P1, c)
                                        + O.mu / O.rho * op.lap(v1, c)
                                        - op.grad(v1, c) * v1[c]
                                        + O.g * (1 - O.beta * (T1[c] - O.T_ref)));
            }

            //Speed of sound
            Range<scal_t> norm = v2.map([](const vec_t& p) { return p.norm(); });
            scal_t C = O.dl * std::max(*std::max_element(norm.begin(), norm.end()), O.v_ref);
            // Mass continuity
            Range<scal_t> div_v;
            for (auto c:all) {
                div_v[c] = op.div(v2, c);
                P2[c] = P1[c] - C * C * O.dt * O.rho * div_v[c] +
                        O.dl2 * C * C * O.dt * O.dt * op.lap(P1, c);
            }
            P1.swap(P2);
        }

        //heat transport
        for (auto c : interior) {
            T2[c] = T1[c] + O.dt * O.lam / O.rho / O.c_p * op.lap(T1, c) -
                    O.dt * v1[c].transpose() * op.grad(T1, c);
        }
        for (auto c : top) T2[c] = op.neumann(T2, c, vec_t{0, -1}, 0.0);
        for (auto c : bottom) T2[c] = op.neumann(T2, c, vec_t{0, 1}, 0.0);
    }

Explicit pressure correction

The solution of heat equation is the same as in above example

    for (int step = 0; step <= O.t_steps; ++step) {
        // Explicit Navier-Stokes computed on whole domain, including boundaries
        // without pressure
        for (int c:all) {
            v_2[c] = v_1[c] + O.dt (
                                   O.mu / O.rho * op.lap(v_1, c)
                                    - op.grad(v_1, c) * v_1[c]
                                    + O.g * (1 - O.beta * (T_1[c] - O.T_ref)));
        }
        // Pressure correction
        VecXd rhs_pressure(N + 1, 0); //Note N+1, +1 stands for regularization equation
        rhs_pressure(N) = 0; // = 0 part of the regularization equation
        for (int i:interior) rhs_pressure(c) = O.rho / O.dt * op.div(v_2, c);
        for (int i: boundary) rhs_pressure(c) = O.rho / O.dt * v_2[c].dot(domain.normal(c));
        VecXd solution = solver_p.solve(rhs_pressure);
        alpha = solution[N];
        VecXd P_c = solution.head(N);
        for ( int i = interior) v_2[c] -=  O.dt / O.rho * op.grad(P_c, c);
        v_2[boundary] = 0; // force boundary conditions
        //heat transport
        for (auto c : interior) {
            T2[c] = T1[c] + O.dt * O.lam / O.rho / O.c_p * op.lap(T1, c) -
                    O.dt * v1[c].transpose() * op.grad(T1, c);
        }
        for (auto c : top) T2[c] = op.neumann(T2, c, vec_t{0, -1}, 0.0);
        for (auto c : bottom) T2[c] = op.neumann(T2, c, vec_t{0, 1}, 0.0);
}

Implicit

for (int step = 0; step <= O.t_steps_i; ++step) {
        time_1 = std::chrono::high_resolution_clock::now();
        // NAVIER STOKES
        M_velocity = mat_t(2 * N, 2 * N);
        // system
        M_velocity.reserve(Range<int>(2 * N, O.n));
        for (int i : interior) {
            op.valuevec(M_velocity, i, 1 / O.dt_i);
            op.lapvec(M_velocity, i, -O.mu / O.rho);
            op.gradvec(M_velocity, i, v_1[i]);
        }
        for (int i : boundary) op.valuevec(M_velocity, i, 1); //sets velocity to 0

        M_velocity.makeCompressed();
        solver_v.compute(M_velocity);
        // solution
        Range <vec_t> rhs_vec(domain.size(), 0);
        for (int i : interior) {
            rhs_vec[i] = -1 / O.rho * op.grad(P, i) +
                         v_1[i] / O.dt_i
                         + O.g * (1 - O.beta * (T[i] - O.T_ref));
        }
        // for (int i:top) rhs_vec[i] = vec_t{0,1};

        v_2 = reshape<2>(solver_v.solveWithGuess(reshape(rhs_vec), reshape(v_1)));
        // END OF NAVIER STOKES

        // PRESSURE CORRECTION
        VecXd rhs_pressure(N + 1, 0); //Note N+1, +1 stands for regularization equation
        rhs_pressure(N) = 0; // = 0 part of the regularization equation
        double alpha;
        for (int i : interior) rhs_pressure(i) = O.rho / O.dt_i * op.div(v_2, i);
        for (int i : boundary) rhs_pressure(i) = O.rho / O.dt * v_2[i].dot(domain.normal(i));

        VecXd solution = solver_p.solve(rhs_pressure);
        alpha = solution[N];
        VecXd P_c = solution.head(N);
        // apply velocity correction
        for (int i : interior) {
            v_2[i] -= O.dl * O.dt_i / O.rho * op.grad(P_c, i);
        }
        P += O.dl * P_c;
        // enforce velocity BC
        // v_2[boundary] = 0;
        // END OF PRESSURE CORRECTION

        // HEAT TRANSPORT
        M_temperature = mat_t(N, N);
        Range<int> per_row(N, O.n);
        M_temperature.reserve(per_row);
        // outer boundary dirichlet BC
        for (int i : top) op.neumann_implicit(M_temperature, i, domain.normal(i), 1);
        for (int i : bottom) op.neumann_implicit(M_temperature, i, domain.normal(i), 1);
        for (int i : left) op.value(M_temperature, i, 1.0);
        for (int i : right) op.value(M_temperature, i, 1.0);
        // heat transport in air
        for (int i : interior) {
            op.value(M_temperature, i, 1.0);                          // time dependency
            op.lap(M_temperature, i, -O.dt_i * O.lam / O.rho / O.c_p);  //laplace in interior
            op.grad(M_temperature, i, O.dt * v_2[i]);
        }
        M_temperature.makeCompressed();
        solver_T.compute(M_temperature);

        VectorXd rhs = VectorXd::Zero(N);
        for (int i : interior) rhs(i) = T(i);
        for (int i : top) rhs(i) = 0;
        for (int i : bottom) rhs(i) = 0;
        for (int i : left) rhs(i) = O.T_hot;
        for (int i : right) rhs(i) = O.T_cold;
        T = solver_T.solveWithGuess(rhs, T);
        // END OF HEAT TRANSPORT
        v_1.swap(v_2);
}

Results

Comparison of MLSM solution with reference data

Following video shows evolution of temperature and velocity magnitude for the $Ra=10^8$ case.

In below galley you can find temperature contour plots, velocity magnitude contour plots, v_max and average hot side Nusselt number convergence behavior. The reference values are from:

[a] de Vahl Davis G. Natural convection of air in a square cavity: a bench mark numerical solution. Int J Numer Meth Fl. 1983;3:249-64.
[b] Sadat H, Couturier S. Performance and accuracy of a meshless method for laminar natural convection. Numer Heat Transfer. 2000;B37:455-67.
[c] Wan DC, Patnaik BSV, Wei GW. A new benchmark quality solution for the buoyancy-driven cavity by discrete singular convolution. Numer Heat Transfer. 2001;B40:199-228.
[d] Šarler B. A radial basis function collocation approach in computational fluid dynamics. CMES-Comp Model Eng. 2005;7:185-93.
[e] Kosec G, Šarler B. Solution of thermo-fluid problems by collocation with local pressure correction. Int J Numer Method H. 2008;18:868-82.