Energy equation

The instantaneous total energy equation describes the conservation of energy in fluid dynamics, accounting for internal energy, velocity, heat flux, and heat generation.

The instantaneous total energy equation [1], in tensorial notation is:

where:

  • e is the internal energy.
  • U is the velocity magnitude.
  • qi is the heat flux in the direction xi.
  • q' is a heat generation or heat sink per unit volume.

In energy conservation equation, the terms represent, in order:

  • The rate of energy gain per unit volume
  • The rate of energy input per unit volume due to convection
  • The rate of energy addition due to conduction
  • The rate at which work is done on the fluid by pressure
  • The rate of energy addition due to viscous forces (dissipation term)
  • The rate of heat generation by internal sources

The second and fourth terms in the above equation can be combined and using the definition of the total enthalpy, ho as:

where h is the static enthalpy of the fluid and combining with total energy conservation equation, gives:

where Sh is the energy source term.

High speed equation

After modeling the conduction and taking the Reynolds average of energy conservation equation [1], it becomes:

This equation expresses the conservation of the total energy for example, the thermal energy plus the mechanical energy. It is valid for all flow situations, but you should limit it to supersonic flow regimes. Because the flow solver is not density based, hypersonic flows are not supported.

Low speed equation

For low speed flows (Mach < 0.3), the energy equation is simplified for robustness and for round-off purposes. For low speed incompressible and compressible flows, the pressure work and dissipation terms in energy conservation equation are neglected. The simplified form of the energy equation has the mechanical energy subtracted from the total energy, and becomes a thermal energy equation [2]:

where:

  • k is the thermal conductivity.
  • h' is the fluctuating static enthalpy.
  • is the turbulent or Reynolds flux.

This form of the equation ensures conservation of the thermal energy, and avoids round-off problems. The low speed form of the energy equation is used by default in the flow solver.

The transient term in the low-speed form of the energy equation should represent storage of internal energy, however, by default, it is implemented as a storage of enthalpy. In transient cases where the pressure changes in time, such as tank charging problems with ideal gas fluids, this approximation may lead to unacceptable errors. You can include pressure time derivative, , to the right side of the energy equation by using the INCLUDE DP/DT TERM IN LOW-SPEED ENERGY EQUATION advanced parameter, so that the correct energy storage rate is recovered.

The viscous heating term (dissipation term) can be activated for the low speed energy equation by using the INCLUDE_VISCOUS_HEATING advanced parameter. This term should be included to the energy equation for the flows with high velocity gradients and incompressible flows. The term is computed on nodes based on the contributions of all surrounding nodes. In addition, a correction is added to account for the different flux calculation across a wall boundary. In this case this wall flux value is added to the existing value of nodes on wall boundaries.