Scale-Resolving Hybrid (SRH) Model

The Scale-Resolving Hybrid (SRH) turbulence model is a continuous hybrid RANS-LES model that combines the accurate prediction capabilities of LES with the low numerical cost of RANS.

On fine meshes and when the time step is small, the SRH approach allows a RANS model to continuously switch to LES mode and resolve unsteady information of large-scale turbulent structures. In Simcenter STAR-CCM+, this hybrid method can be applied to K-Epsilon and K-Omega models.

The equations that are solved for the SRH approach are obtained by a spatial-temporal filtering. Each solution variable $ϕ$ is decomposed into a filtered value $\bar{\tilde{ϕ}}$ and a subfiltered value $ϕ'$ :

Figure 1. EQUATION_DISPLAY

ϕ = \bar{\tilde{ϕ}} + ϕ'

(1294)

where $ϕ$ represents velocity components, pressure, energy, or species concentration.

The SRH filter ensures that for large filter time-width, the transport equations are equivalent to the RANS equations, and for short filter-width, the equations are equivalent to LES. Between these time scales, the time step and the mesh size determine the resolution of the turbulent structures. For more background information, see [337].

Inserting the decomposed solution variables into the Navier-Stokes equations results in filtered transport equations for mass, momentum, and energy that look identical to the transport equations that are solved for RANS and LES simulations. However, the stress tensor $T_{S F S}$ now represents the subfiltered scale stresses.

These stresses are modeled using the Boussinesq approximation:

Figure 2. EQUATION_DISPLAY

T_{S F S} = 2 μ_{t} S - \frac{2}{3} ρ k_{S F S} I

(1295)

where:

$S$ is the strain rate tensor given by Eqn. (1130) and computed from the resolved velocity field $\bar{\tilde{v}}$ .
$ρ$ is the density.
$k_{S F S}$ is the subfilter scale turbulent kinetic energy.
$I$ is the identity tensor.

The turbulent eddy viscosity $μ_{t}$ is defined by the respective RANS model and is a function of a turbulent time scale and the turbulent kinetic energy. For the SRH approach the subfilter scale turbulent kinetic energy $k_{S F S}$ is given by the solution of the following transport equation ([337]):


RANS Model	Transport Equation for $k_{S F S}$
K-Epsilon	Figure 3. EQUATION_DISPLAY $\frac{\partial}{\partial t} (ρ k_{S F S}) + \nabla\cdot (ρ k_{S F S} \bar{\tilde{v}}) = \nabla\cdot [(μ + \frac{μ_{t}}{σ_{k}}) \nabla k_{S F S}] + P_{k_{S F S}} - ρ {(ψ}_{H} ε - ε_{0}) + S_{k_{S F S}}$ (1296)
K-Omega	Figure 4. EQUATION_DISPLAY $\frac{\partial}{\partial t} (ρ k_{S F S}) + \nabla\cdot (ρ k_{S F S} \bar{\tilde{v}}) = \nabla\cdot [(μ + σ_{k} μ_{t}) \nabla k_{S F S}] + P_{k_{S F S}} - ρ β^{} f_{β^{}} (ψ_{H} ω k_{S F S} - ω_{0} k_{0}) + S_{k_{S F S}}$ (1297)

where:

$μ$ is the dynamic viscosity.
$σ_{k}$ and $β^{*}$ are model coefficients of the respective RANS model.
$P_{k_{S F S}}$ is the production term for the subfilter-stress turbulent kinetic energy as defined by the respective RANS model.
$ε$ is the turbulent dissipation rate.
$ε_{0}$ is the ambient turbulence value in the source terms that counteracts turbulence decay [316].
$f_{β^{*}}$ is the free-shear modification factor.
$ω$ is the specific turbulent dissipation rate.
$S_{k_{S F S}}$ is the user-specified source term.

The parameter $ψ_{H}$ is calculated as:

Figure 5. EQUATION_DISPLAY

ψ_{H} = 1 - f_{s} + f_{s} \frac{1}{r_{k}} (\frac{4}{3} - \frac{1}{3} r_{k}^{3 / 4}) \frac{k_{S F S}}{k_{m} + k_{r}}

(1298)

where:

$f_{s}$ is a shielding function, which imposes the RANS mode in the near-wall regions, defined as:

Figure 6. EQUATION_DISPLAY

$f_{s} = 1 - \tanh [{(r_{d c} \frac{ν^{3 / 4}}{ε^{1 / 4} d})}^{8}]$

(1299)
where:
- $r_{d c}$ is a Model Coefficient that controls the thickness of the shielding layer.
- $ν$ is the kinematic viscosity.
- $d$ is the wall distance.
$k_{m}$ and $k_{r}$ are the time-averaged values of the subfilter scale modeled turbulent kinetic energy and the resolved kinetic energy, respectively, and defined as:

Figure 7. EQUATION_DISPLAY

$k_{m} = \bar{k_{S F S}}$

(1300)

Figure 8. EQUATION_DISPLAY

$k_{r} = \frac{1}{2} (\bar{\tilde{v} \cdot \tilde{v}} - \bar{\tilde{v}} \cdot \bar{\tilde{v}})$

(1301)

Figure 9. EQUATION_DISPLAY

$k_{t} = k_{m} + k_{r}$

(1302)
$r_{k}$ is the theoretical energy ratio defined as:

Figure 10. EQUATION_DISPLAY

$r_{k} = 1 - \max (0, 1 - β_{ψ} {C F L}_{H}^{2 / 3})$

(1303)

where $β_{ψ}$ is a Model Coefficient that controls the effectiveness of the scaling parameter $ψ_{H}$ .
The CFL number of the filtered quantities ${C F L}_{H}$ is calculated as:
Figure 11. EQUATION_DISPLAY

${C F L}_{H} = \frac{ε}{k_{t}^{3 / 2}} \max (Δ t U_{s}, Δ)$

(1304)
where:
- $Δ t$ is the time-step size.
- $Δ$ is a local length-scale.
$U_{s}$ is the sweeping velocity of the turbulence structures defined as:
Figure 12. EQUATION_DISPLAY

$U_{s} = | \bar{\tilde{v}} | + \sqrt{k_{t}}$

(1305)

Finally, the shielding function $f_{s}$ is used to define the blending factor for the hybrid upwind differencing schemes for the convective flux as:

Figure 13. EQUATION_DISPLAY

σ_{H U} = 1 - f_{s}

(1306)

Model Coefficients


$r_{d c}$	$β_{ψ}$
45	0.69