Closure conditions in the form of equation of state

Question

While reading the book "Riemann solvers and numerical methods for fluid dynamics" By E. Toro, the very first paragraph is:

"In this chapter, we present the governing equations for the dynamics of a compressible material, such as a gas, along with closure conditions in the form of equations of state. Equations of state are statements about the nature of the material in question and require some notions from Thermodynamics. There is no attempt to provide an exhaustive and rigourous derivation of the equations of continuum mechanics; such a task is beyond the scope of this book. Instead, we give a fairly self–contained summary of the equations and the Thermodynamics in a manner that is immediately useful to the main purpose of this book, namely the detailed treatment of Riemann solvers and numerical methods"

What is meant by "closure condition"? I have been searching, and I think this means that the number of variables in the set of equations for dynamics of compressible material is more than the number of equations. Thus we need additional equations. But I am not sure about this.

Answer 1

The Eulerian hydrodynamic equations are given in terms of the fluid density $\rho$, the fluid velocity $\mathbf u$ and pressure $p$ (or sometimes total energy $E$). This means there are between 3 and 5 variables, depending on the dimension of the problem, but you only have 2 to 4 equations to work with (1 from mass continuity, 1 to 3 from momentum conservation equation). Thus, we need an additional equation to close out the full system of equations so that there is the same number of unknowns as equations (which allows it to be solved without making other assumptions).

Often, the pressure equation takes the form, $$ \frac{\partial p}{\partial t}+\mathbf u\cdot\nabla p+\gamma p\nabla\cdot\mathbf u=S $$ where $S$ is the source/sink term (i.e., energy production and heat sink), which we often take to be zero, and $\gamma$ the adiabatic index. In Toro's book, however, you are more likely to see the third equation of hydrodynamics written in terms of the total energy, $E=\frac12\rho \mathbf u\cdot\mathbf u+p\left(\gamma-1\right)^{-1}$, which gives you the conservation equation, $$ \frac{\partial E}{\partial t}+\nabla\cdot\left[\left(E+p\right)\mathbf u\right]=S $$ In some rare cases, if you are working with an ideal gas, you can actually define the closure condition in terms of entropy, as I indicate in this other answer of mine (note I ignore sources there).

Answer 2

Generally one is expected to have as many equations as one has unknowns. In this sense the equations of elasticity theory or fluid dynamics are usually not complete, and require using thermodynamic equations to have meaningful solutions. This is what here by closure - closing the system of equations (admittedly a convoluted way of stating it).