Consequences of the second law for a turbulent fluid flow

Second Law statements in thermomechanics applicable to turbulent fluid flow, in which the internal energy in a macroscopic field theory includes contributions both from molecular vibrations and from turbulent fluctuations, are discussed. In the absence of turbulence, these statements naturally reduce to the known and accepted Second Law statements for a nonturbulent medium. The usual version of the Second Law statements — which deny the existence of perpetual motion and place restrictions on the constitutive equations —is extended here in the presence of turbulence; and an additional statement is introduced associated with the tendency of turbulent fluctuations to decay in the absence of external work or the addition of thermal heat. The mathematical representations of various Second Law statements are then used to derive several restrictions on the response variables of the macroscopic turbulence theory. Examples of such variables include the rates of production and dissipation of turbulent fluctuations, the rate of thermal entropy production, internal energy (involving constitutive coefficients which may be taken to be the thermal and turbulent specific heats), turbulent viscosity coefficients and other response functions which control the degree of flow anisotropy in the medium. These Second Law restrictions are then applied to a recent theory of macroscopic turbulent flow by the present authors in which fairly general constitutive equations are presented for the dependent variables of the theory. It is found that not only is the range of values of several constitutive coefficients limited by these Second Law restrictions, but the presence of a number of terms in the constitutive equations is entirely denied.