1 // Copyright (c) 2017-2022, Lawrence Livermore National Security, LLC and other CEED contributors. 2 // All Rights Reserved. See the top-level LICENSE and NOTICE files for details. 3 // 4 // SPDX-License-Identifier: BSD-2-Clause 5 // 6 // This file is part of CEED: http://github.com/ceed 7 8 /// @file 9 /// Density current initial condition and operator for Navier-Stokes example using PETSc 10 11 // Model from: 12 // Semi-Implicit Formulations of the Navier-Stokes Equations: Application to 13 // Nonhydrostatic Atmospheric Modeling, Giraldo, Restelli, and Lauter (2010). 14 15 #ifndef densitycurrent_h 16 #define densitycurrent_h 17 18 #include <math.h> 19 #include <ceed.h> 20 #include "newtonian_types.h" 21 22 #ifndef M_PI 23 #define M_PI 3.14159265358979323846 24 #endif 25 26 // ***************************************************************************** 27 // This function sets the initial conditions and the boundary conditions 28 // 29 // These initial conditions are given in terms of potential temperature and 30 // Exner pressure and then converted to density and total energy. 31 // Initial momentum density is zero. 32 // 33 // Initial Conditions: 34 // Potential Temperature: 35 // theta = thetabar + delta_theta 36 // thetabar = theta0 exp( N**2 z / g ) 37 // delta_theta = r <= rc : thetaC(1 + cos(pi r/rc)) / 2 38 // r > rc : 0 39 // r = sqrt( (x - xc)**2 + (y - yc)**2 + (z - zc)**2 ) 40 // with (xc,yc,zc) center of domain, rc characteristic radius of thermal bubble 41 // Exner Pressure: 42 // Pi = Pibar + deltaPi 43 // Pibar = 1. + g**2 (exp( - N**2 z / g ) - 1) / (cp theta0 N**2) 44 // deltaPi = 0 (hydrostatic balance) 45 // Velocity/Momentum Density: 46 // Ui = ui = 0 47 // 48 // Conversion to Conserved Variables: 49 // rho = P0 Pi**(cv/Rd) / (Rd theta) 50 // E = rho (cv T + (u u)/2 + g z) 51 // 52 // Boundary Conditions: 53 // Mass Density: 54 // 0.0 flux 55 // Momentum Density: 56 // 0.0 57 // Energy Density: 58 // 0.0 flux 59 // 60 // Constants: 61 // theta0 , Potential temperature constant 62 // thetaC , Potential temperature perturbation 63 // P0 , Pressure at the surface 64 // N , Brunt-Vaisala frequency 65 // cv , Specific heat, constant volume 66 // cp , Specific heat, constant pressure 67 // Rd = cp - cv, Specific heat difference 68 // g , Gravity 69 // rc , Characteristic radius of thermal bubble 70 // center , Location of bubble center 71 // dc_axis , Axis of density current cylindrical anomaly, or {0,0,0} for spherically symmetric 72 // ***************************************************************************** 73 74 // ***************************************************************************** 75 // This helper function provides support for the exact, time-dependent solution 76 // (currently not implemented) and IC formulation for density current 77 // ***************************************************************************** 78 CEED_QFUNCTION_HELPER int Exact_DC(CeedInt dim, CeedScalar time, 79 const CeedScalar X[], CeedInt Nf, CeedScalar q[], 80 void *ctx) { 81 // Context 82 const SetupContext context = (SetupContext)ctx; 83 const CeedScalar theta0 = context->theta0; 84 const CeedScalar thetaC = context->thetaC; 85 const CeedScalar P0 = context->P0; 86 const CeedScalar N = context->N; 87 const CeedScalar cv = context->cv; 88 const CeedScalar cp = context->cp; 89 const CeedScalar *g_vec = context->g; 90 const CeedScalar rc = context->rc; 91 const CeedScalar *center = context->center; 92 const CeedScalar *dc_axis = context->dc_axis; 93 const CeedScalar Rd = cp - cv; 94 const CeedScalar g = -g_vec[2]; 95 96 // Setup 97 // -- Coordinates 98 const CeedScalar x = X[0]; 99 const CeedScalar y = X[1]; 100 const CeedScalar z = X[2]; 101 102 // -- Potential temperature, density current 103 CeedScalar rr[3] = {x - center[0], y - center[1], z - center[2]}; 104 // (I - q q^T) r: distance from dc_axis (or from center if dc_axis is the zero vector) 105 for (CeedInt i=0; i<3; i++) 106 rr[i] -= dc_axis[i] * 107 (dc_axis[0]*rr[0] + dc_axis[1]*rr[1] + dc_axis[2]*rr[2]); 108 const CeedScalar r = sqrt(rr[0]*rr[0] + rr[1]*rr[1] + rr[2]*rr[2]); 109 const CeedScalar delta_theta = r <= rc ? thetaC*(1. + cos(M_PI*r/rc))/2. : 0.; 110 const CeedScalar theta = theta0*exp(N*N*z/g) + delta_theta; 111 112 // -- Exner pressure, hydrostatic balance 113 const CeedScalar Pi = 1. + g*g*(exp(-N*N*z/g) - 1.) / (cp*theta0*N*N); 114 // -- Density 115 116 const CeedScalar rho = P0 * pow(Pi, cv/Rd) / (Rd*theta); 117 118 // Initial Conditions 119 q[0] = rho; 120 q[1] = 0.0; 121 q[2] = 0.0; 122 q[3] = 0.0; 123 q[4] = rho * (cv*theta*Pi + g*z); 124 125 return 0; 126 } 127 128 // ***************************************************************************** 129 // This QFunction sets the initial conditions for density current 130 // ***************************************************************************** 131 CEED_QFUNCTION(ICsDC)(void *ctx, CeedInt Q, 132 const CeedScalar *const *in, CeedScalar *const *out) { 133 // Inputs 134 const CeedScalar (*X)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[0]; 135 136 // Outputs 137 CeedScalar (*q0)[CEED_Q_VLA] = (CeedScalar(*)[CEED_Q_VLA])out[0]; 138 139 CeedPragmaSIMD 140 // Quadrature Point Loop 141 for (CeedInt i=0; i<Q; i++) { 142 const CeedScalar x[] = {X[0][i], X[1][i], X[2][i]}; 143 CeedScalar q[5] = {0.}; 144 145 Exact_DC(3, 0., x, 5, q, ctx); 146 147 for (CeedInt j=0; j<5; j++) 148 q0[j][i] = q[j]; 149 } // End of Quadrature Point Loop 150 151 return 0; 152 } 153 154 #endif // densitycurrent_h 155