Entropy stable boundary conditions are critical for ensuring that the corresponding numerical scheme satisfies the discrete entropy inequality, thus, mimicking the second law of thermodynamics for the compressible Navier-Stokes equations. Provably entropy-stable adiabatic wall boundary conditions were introduced in Parsani 2014. These discrete wall boundary conditions were further generalized in Dalcin 2019 to a moving adiabatic solid wall or a wall with a prescribed heat flux for the compressible Navier--Stokes equations discretized by using summation-by-parts (SBP) and simultaneous-approximation-term (SAT) operators. Also, a similar approach was used in Chan 2022 to impose adiabatic and isothermal no-slip wall boundary conditions for the modal Discontinuous Galerkin method for the compressible Navier-Stokes equations. Although the adiabatic wall boundary conditions developed in Chan 2022 are provably stable in the entropy sense, the authors report that while the proposed discrete isothermal no-slip wall boundary conditions mimic the continuous entropy inequality, they are not entropy-stable and invariably lead to positive entropy production. Moreover, to the best of our knowledge, no entropy-stable formulation for isothermal wall boundary conditions is currently available in the literature. This talk presents a new isothermal no-slip wall boundary conditions that: 1) enforce the T|_{y=0}=T^wall and v|_{y=0}=v^{wall} conditions with the design order of accuracy while maintaining stability, 2) provide the correct sign of entropy production at the wall, and 3) mimic the entropy balance of the Navier-Stokes equations at the discrete level. The structure-preserving and design-order properties of the proposed methodology will be demonstrated and verified on standard benchmark problems for compressible flows. References: - M. Parsani, M. H. Carpenter, and E. J. Nielsen, “Entropy stable wall boundary conditions for the compressible Navier--Stokes equations,” tech. rep., 2014. - L. Dalcin, D. Rojas, S. Zampini, D. C. D. R. Fernández, M. H. Carpenter, and M. Parsani, “Conservative and entropy stable solid wall boundary conditions for the compressible Navier--Stokes equations: Adiabatic wall and heat entropy transfer,” JCP, vol. 397, p. 108775, 2019. - J. Chan, Y. Lin, and T. Warburton, “Entropy stable modal discontinuous Galerkin schemes and wall boundary conditions for the compressible Navier--Stokes equations,” JCP, vol. 448, p. 110723, 2022.