A review of the diverse roles of entropy and the second law in computationalthermo-uid dynamics is presented. Entropy computations are related to numerical error, convergence criteria, time-step limitations, and other signi cant aspects of computational uid ow and heat transfer. The importance of the second law as a tool for estimating error bounds and the overall scheme's robustness is described. As computational methods become more reliable and accurate, emerging applications involving the second law in the design of engineering thermal uid systems are described. Sample numerical results are presented and discussed for a multitude of applications in compressible ows, as well as problems with phase change heat transfer. Advantages and disadvantages of different entropy-based methods are discussed, as well as areas of importance suggested for future research. Nomenclature c p = speci c heat, J/kg ¢ K E = total energy density, J/m 3 F = entropy ux F F = ux column vector I S = entropy current k = thermal conductivity, W/m ¢ K P = pressure, N/m 3 Q Q = vector of conserved variables q = heat ux, W/m 2 R = gas constant per unit mass, m 2 /s 2 ¢ K S = entropy density (volumetric), J/m 3 ¢ K P S gen = rate of entropy generation, W/m 3 ¢ K s = speci c entropy, J/kg ¢ K T = absolute temperature, K t = time, s U = internal energy density, J/m 3 v = uid velocity vector, m/s x, y = Cartesian coordinates, m°= ratio of speci c heats 1 = increment or change of quantity ¹ = dynamic viscosity, kg/m ¢ s ½ = mass density, kg/m 3 ¿ = viscous stress, N/m 2 8 = viscous dissipation function, 1/s 2
