Physics related keywords

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Compressible flow
- Shock capturing
  - Spectral vanishing viscosity

Compressible flow

Keyword	Description	Default value
Mach number	REAL:	Mandatory keyword
Reynolds number	REAL:	Mandatory keyword
Prandtl number	REAL:	0.72
Turbulent Prandtl number	REAL:	Equal to Prandtl
AOA theta	REAL: Angle of attack (degrees), based on the spherical coordinates polar angle ( $\theta$ ) definition	0.0
AOA phi	REAL: Angle of attack (degrees), based on the spherical coordinates azimuthal angle ( $\varphi$ ) definition	0.0
LES model	CHARACTER(*): Options are: Vreman, Wale, Smagorinsky, None	None
Wall model	CHARACTER:	linear

Shock capturing

The shock-capturing module helps stabilize cases with discontinuous solutions, and may also improve the results of under-resolved turbulent cases. It is built on top of a Sensors module that detects problematic flow regions, classifying them according to the value of the sensor, $s$ , mapped into the interval $a \in [0,1]$ ,

$a = \left\{\begin{array}{ll} 0, & \text{if } s \leq s_0 - \Delta s / 2, \\ \frac{1}{2}\left[1+\sin\left(\frac{s-s_0}{\Delta s}\right)\right], & \text{if } s_0 - \Delta s / 2 < s < s_0 + \Delta s / 2, \\ 1, & \text{elsewhere}. \end{array}\right.$

The values of $s_0 = (s_1 + s_2)/2$ and $\Delta s = s_2 - s_1$ depend on the sensor thresholds $s_1$ and $s_2$ .

At the moment, flow regions where $a \leq 0$ are considered smooth and no stabilization algorithm can be imposed there. In the central region of the sensor, with $0 < a < 1$ , the methods shown in the next table can be used and even scaled with the sensor value, so that their intensity increases in elements with more instabilities. Finally, the higher part of the sensor range can implement a different method from the table; however, the intensity is set to the maximum this time.

All the methods implemented introduce artificial dissipation into the equations, which can be filtered with an SVV kernel to reduce the negative impact on the accuracy of the solution. Its intensity is controlled with the parameters $\mu$ (similar to the viscosity of the Navier-Stokes equations) and $\alpha$ (scaling of the density-regularization term of the Guermond-Popov flux), which can be set as constants or coupled to the value of the sensor or to a Smagorinsky formulation.

Keyword	Description	Default value
Enable shock-capturing	LOGICAL: Switch on/off the shock-capturing stabilization	.FALSE.
Shock sensor	CHARACTER: Type of sensor to be used to detect discontinuous regions Options are: Zeros: always return 0 Ones: always return 1 Integral: integral of the sensor variable inside each element Integral with sqrt: square root of the Integral sensor Modal: based on the relative weight of the higher order modes Truncation error: estimate the truncation error of the approximation Aliasing error: estimate the aliasing error of the approximation GMM: clustering sensor based on the divergence of the velocity and the gradient of the pressure	Integral
Shock first method	CHARACTER: Method to be used in the middle region of the sensor ( $a\in[0,1]$ ). Options are: None: Do not apply any smoothing Non-filtered: Apply the selected viscous flux without SVV filtering SVV: Apply an entropy-stable, SVV-filtered viscous flux	None
Shock second method	CHARACTER: Method to be used in the top-most region of the sensor ( $a=1$ ). Options are: None: Do not apply any smoothing Non-filtered: Apply the selected viscous flux without SVV filtering SVV: Apply an entropy-stable, SVV-filtered viscous flux	None
Shock viscous flux 1	CHARACTER: Viscous flux to be applied in the elements where $a\in[0,1]$ . Options are: Physical Guermond-Popov (only with entropy variables gradients)	--
Shock viscous flux 2	CHARACTER: Viscous flux to be applied in the elements where $a=1$ . Options are: Physical Guermond-Popov (only with entropy variables gradients)	--
Shock update strategy	CHARACTER: Method to compute the variable parameter of the specified shock-capturing approach in the middle region of the sensor. Options are: Constant Sensor Smagorinsky: only for non-filtered and SVV	Constant
Shock mu 1	REAL/CHARACTER()*: Viscosity parameter $\mu_1$ , or $C_s$ in the case of LES coupling	0.0
Shock alpha 1	REAL: Viscosity parameter $\alpha_1$	0.0
Shock mu 2	REAL: Viscosity parameter $\mu_2$	$\mu_1$
Shock alpha 2	REAL: Viscosity parameter $\alpha_2$	$\alpha_1$
Shock alpha/mu	REAL: Ratio between $\alpha$ and $\mu$ . It can be specified instead of $\alpha$ itself to make it dependent on the corresponding values of $\mu$ , and it is compulsory when using LES coupling	--
SVV filter cutoff	REAL/CHARACTER()*: Cutoff of the filter kernel, $P$ . If "automatic", its value is adjusted automatically	"automatic"
SVV filter shape	CHARACTER()*: Options are: Power Sharp Exponential	Power
SVV filter type	CHARACTER()*: Options are: Low-pass High-pass	High-pass
Sensor variable	CHARACTER()*: Variable used by the sensor to detect shocks. Options are: rho rhou rhov rhow u v w p rhop grad rho div v	rhop
Sensor lower limit	REAL: Lower threshold of the central sensor region, $s_1$	Mandatory keyword (except GMM)
Sensor higher limit	REAL: Upper threshold of the central sensor region, $s_2$
Sensor TE min N	INTEGER: Minimum polynomial order of the coarse mesh used for the truncation error estimation	1
Sensor TE delta N	Polynomial order difference between the solution mesh and its coarser representation	1
Sensor TE derivative	CHARACTER: Whether the face terms must be considered in the estimation of the truncation error or not. Options are: Non-isolated Isolated	Isolated
Sensor number of clusters	INTEGER: Maximum number of clusters to use with the GMM sensor	2
Sensor min. steps	INTEGER: Minimum number of time steps that an element will remain detected. The last positive value will be used if the sensor "undetects" an element too early	1

Spectral vanishing viscosity

The introduction of an SVV-filtered artificial flux helps dissipate high-frequency oscillations. The baseline viscous flux can be chosen as the Navier-Stokes viscous flux or the flux developed by Guermond and Popov. In any case, this flux is expressed in a modal base where it is filtered by any of the following three filter kernels:

power: $\hat{F}^{\text{1D}}_i = (i/N)^P$ ,
sharp: $\hat{F}^{\text{1D}}_i = 0$ if (i