Energy equations

Energy equations#

The energy equations are usually required for compressible flows when thermodynamic changes (e.g. changes in enthalpy or temperature) in the flow field become relevant, such as flows featuring acoustics or combustion/chemical reactions. Various forms of the energy equation exist, each tailored to specific physical problems and the assumptions made during their derivation.

References:

Nomenclature:

  • \(\mathbf{u}\): velocity vector

  • \(p\): pressure

  • \(T\): temperature

  • \(T_b\): burnt, adiabatic flame temperature

  • \(T_u\): unburnt temperature

  • \(\rho\): density

  • \(c_p\): heat capacity at constant pressure

  • \(\gamma\): heat capacity ratio

  • \(\kappa\): thermal conductivity

  • \(\mathbf{\tau}\): viscous stress tensor

  • \(\mu\): dynamic viscosity

  • \(c\): progress variable

  • \(\mathbf{I}\): identity tensor

  • \(X_h\): test function for energy equation

Two forms of energy equations are currently implemented in FELiCS.

1. Energy pressure equation#

This form is based on the conservation of sensible energy. By assuming a heat capacity independent from temperature, one can obtain an equation for pressure. It is used here mainly for compressible non-reacting flows. For reference, this form of the energy equation corresponds to eq. (1.75) in the book “Theoretical and Numerical Combustion” by Poinsot & Veynante, without chemical reaction terms.

Assumptions:

  • primitive variables

  • constant Prandtl number: \(Pr=\frac{\mu c_p}{\kappa}=cst\)

  • constant heat capacity ratio

  • heat capacity independent from temperature

  • molecular viscosity adheres to Sutherland’s law

  • fluctuations in thermal conductivity are tied to the given Prandtl number and, thus, eddy viscosity

Example use case:

Nonlinear energy pressure equation#

The nonlinear energy pressure equation is

\[ \frac{\partial p}{\partial t} + \mathbf{u} \cdot \nabla p + \gamma p(\nabla \cdot \mathbf{u}) - (\gamma - 1)\left[ \nabla (\kappa \nabla T) + \tau : \nabla \mathbf{u}\right] = 0 \]

with \(\mu_{mol}\) the molecular viscosity and the viscous stress tensor \(\tau\) being defined as

\[ \tau = \mu \left[\left(\nabla + \nabla ^T\right)\mathbf{u} - \frac{2}{3} (\nabla \cdot \mathbf{u}) \mathbf{I}\right]. \]

Note

The nonlinear energy pressure equation is currently not implemented in FELiCS.

Linearized energy pressure equation#

The linear energy pressure equation is

\[ \frac{\partial p'}{\partial t} + \overline{\mathbf{u}} \cdot \nabla p' + \mathbf{u}' \cdot \nabla \overline{p} + \overline{\gamma}\left[ \overline{p}(\nabla \cdot \mathbf{u}') + p'(\nabla \cdot \overline{\mathbf{u}}) \right] - (\overline{\gamma} - 1)\left[ \nabla (\overline{\kappa} \nabla T' + \kappa' \nabla \overline{T} + \overline{\tau} : \nabla \mathbf{u}' + \tau' : \nabla \overline{\mathbf{u}}) \right] = 0, \]

where the mean viscous stress tensor \(\overline{\tau}\) is defined as

\[ \mathbf{\overline{\tau}} = \overline{\mu}_\textrm{eff}[(\nabla + \nabla ^T)\overline{\mathbf{u}} - \frac{2}{3} (\nabla \cdot \overline{\mathbf{u}}) \mathbf{I}]. \]

The fluctuating viscous shear stress tensor \(\tau'\) is defined as

\[ \tau' = \mu' \left[\left(\nabla + \nabla ^T\right)\overline{\mathbf{u}} - \frac{2}{3} (\nabla \cdot \overline{\mathbf{u}}) \mathbf{I}\right] + \overline{\mu}\left[(\nabla + \nabla ^T) \mathbf{u}' - \frac{2}{3} (\nabla \cdot \mathbf{u}') \mathbf{I}\right] \]

where \(\overline{\mu}_\textrm{eff}\) and \(\mu_\textrm{eff}'\) are the mean and fluctuating effective dynamic viscosity, respectively (see Viscosity models for details).

Weak form of the linearized energy pressure equation#

The weak form of the linearized energy pressure equation with normal mode ansatz, as implemented in FELiCS, is

\[ \int_\Omega \omega \hat{p} \, X_h^* \, \mathrm{d}\mathbf{x} = \int_\Omega \mathrm{j} \, \nabla \cdot \left( X_h^* \, \overline{\mathbf{u}} \right) \, \hat{p} \, \mathrm{d}\mathbf{x} + \int_\Omega \mathrm{j} \, \nabla \cdot \left( X_h^* \, \hat{\mathbf{u}} \right) \, \overline{p} \, \mathrm{d}\mathbf{x} + \int_\Omega \mathrm{j} \, \overline{\gamma} \, \nabla \left( \overline{p} \, X_h^* \right) \cdot \hat{\mathbf{u}} \, \mathrm{d}\mathbf{x} + \int_\Omega \mathrm{j} \, \overline{\gamma} \, \nabla \left( \hat{p} \, X_h^* \right) \cdot \overline{\mathbf{u}} \, \mathrm{d}\mathbf{x} - \int_{\partial\Omega} \mathrm{j} (\overline{\gamma}+1) \, \overline{p} \, \hat{\mathbf{u}} \cdot \mathbf{n} X_h^* \, \mathrm{d}\mathbf{s} - \int_{\partial\Omega} \mathrm{j} (\overline{\gamma}+1) \, \hat{p} \, \overline{\mathbf{u}} \cdot \mathbf{n} X_h^* \, \mathrm{d}\mathbf{s} - \int_\Omega \mathrm{j} (\overline{\gamma} - 1) \, \nabla X_h^* \cdot \left( \overline{\kappa} \nabla \hat{T} \right) \, \mathrm{d}\mathbf{x} + \int_{\partial\Omega} \mathrm{j} (\overline{\gamma} - 1) \, \overline{\kappa} \left( \nabla \hat{T} \cdot \mathbf{n} \right) X_h^* \, \mathrm{d}\mathbf{s} - \int_\Omega \mathrm{j} (\overline{\gamma} - 1) \, \nabla X_h^* \cdot \left( \hat{\kappa} \nabla \overline{T} \right) \, \mathrm{d}\mathbf{x} + \int_{\partial\Omega} \mathrm{j} (\overline{\gamma} - 1) \, \kappa' \left( \nabla \overline{T} \cdot \mathbf{n} \right) X_h^* \, \mathrm{d}\mathbf{s} - \int_\Omega \mathrm{j} (\overline{\gamma} - 1) \, \left(\nabla \cdot \overline{\mathbf{\tau}}\right) \cdot \left( \hat{\mathbf{u}} X_h^* \right) \, \mathrm{d}\mathbf{x} + \int_{\partial\Omega} \mathrm{j} (\overline{\gamma} - 1) \, \left(\overline{\mathbf{\tau}} \hat{\mathbf{u}}\right) \cdot \mathbf{n} X_h^* \, \mathrm{d}\mathbf{s} - \int_\Omega \mathrm{j} (\overline{\gamma} - 1) \, \left( \nabla \cdot \hat{\mathbf{\tau}} \right) \cdot \left( \overline{\mathbf{u}} X_h^* \right) \, \mathrm{d}\mathbf{x} + \int_{\partial\Omega} \mathrm{j} (\overline{\gamma} - 1) \, \left(\hat{\mathbf{\tau}} \overline{\mathbf{u}}\right) \cdot \mathbf{n} X_h^* \, \mathrm{d}\mathbf{s} \]

Note

In practice, the thermal conductivity (mean and fluctuation) is simplified out of the equation by using the Prandtl number: \(\kappa=\frac{\mu_{tot} c_p}{Pr}\).

2. Progress variable equation#

This equation is used to describe an active flame model, which incorporates fluctuations of the reaction rate to capture turbulent flame dynamics more accurately (Kaiser et al. 2023). This model relates the progress variable directly to the reaction rate, enabling a simplified yet effective representation of active flames under turbulent conditions.

Assumptions:

  • progress variable and density are directly related through algebraic equations, which obviates the use of an explicit energy equation \(\rightarrow\) the species equation also acts as an energy conservation equation

  • incompressible flow due to (1) low Mach number, (2) flame being acoustically compact

  • adiabatic walls (no heat losses), adiabatic flame temperature is reached in the entire domain

  • constant heat capacity and constant specific gas constants

  • all assumptions result in temperature only being a function of the progress variable, \(T = T_u + (T_b - T_u)c\)

  • low Mach equation of state, pressure is constant

Example use case:

Nonlinear progress variable equation#

See Species transport equation.

Linearized progress variable equation#

See Species transport equation.

Contents