Compressible Flow: de Laval Nozzle

Overview

The aim of this test case is to validate the following parameters of compressible steady-state turbulent flow through a de Laval Nozzle subsonic and supersonic flow regimes:

Mach number
Pressure

The rhoSimpleFoam solver is used for the subsonic case, while for the supersonic case the sonicFoam solver is employed. The $k - ω S S T$

k - ω S S T

model was used to model turbulence. Simulation results of SimScale were compared to analytical results obtained from methods elucidated in [1]. The mesh was created locally using the blockMesh tool and then imported on to the SimScale platform.

Import validation project into workspace

Geometry

A typical configuration of the de Laval nozzle with a non-smooth throat was chosen as the geometry (see Table 1 for coordinates). Since the nozzle is axisymmetric, it was modeled as an angular slice of the complete geometry, with a wedge angle of 18 degrees (see Fig.1.).

Table 1: Point Coordinates
Point	$x$ $x$	$y$ $y$	$z$ $z$
A	0	0	0
B	0	5.5434	35
C	0	-5.5434	35
D	68.68	0	0
E	68.68	3.1677	20
F	68.68	-3.1677	20
G	238.8	6.3354	40
H	238.8	-6.3354	40

Analysis type and Domain

A non-uniformly-spaced hexahedral mesh was generated using the blockMesh tool (see Fig.2.). Flow near the nozzle wall was resolved using inflation of $y^{+} = 30$

y^{+} = 30

for the subsonic and $300$

300

for the supersonic case. In order to keep the flow two-dimensional, the mesh was designed to have only one layer in the $y$

y

direction.

Tool Type : OPENFOAM®

Analysis Type : Compressible Steady-state (Turbulent)

Mesh and Element types :

Table 2: Mesh Metrics
Mesh type	Cells in x	Cells in y	Cells in z	Number of nodes	Type
blockMesh	150	1	100	15000	2D hex

Simulation Setup

Fluid:

Table 3 encapsulates the properties of fluids used in the subsonic and supersonic case simulations. The need for using a different fluid for supersonic case arises from the courant number restriction.

Table 3: Fluid Properties
Case	$m$ $m$ $g / m o l$ $g / m o l$	$c_{p}$ $c_{p}$ $J / k g K$ $J / k g K$	$m u$ $m u$ $N / m s$ $N / m s$	$P r$ $P r$
Subsonic	$28.9$ $28.9$	$1005$ $1005$	$1.79 \times 10^{- 5}$ $1.79 \times 10^{- 5}$	$1$ $1$

Supersonic	$11640.3$ $11640.3$	$2.5$ $2.5$	$1.8 \times 10^{- 5}$ $1.8 \times 10^{- 5}$	$1$ $1$

Boundary Conditions:

Table 4.1: Boundary Conditions for Subsonic Case
Parameter	Inlet (ABC)	Outlet (GHI)	Wall (AEHIFC)	Wedges (AGHEBA + AGIFCA)
Velocity	$7.58 m s^{- 1}$ $7.58 m s^{- 1}$	Zero Gradient	$0.0 m s^{- 1}$ $0.0 m s^{- 1}$	Wedge
Pressure	Zero Gradient	$1.3 \times 10^{5} N m^{- 2}$ $1.3 \times 10^{5} N m^{- 2}$	Zero Gradient	Wedge
Temperature	$300 K$ $300 K$	Zero Gradient	Zero Gradient	Wedge
$k$ $k$	$0.862 m^{2} / s^{2}$ $0.862 m^{2} / s^{2}$	Zero Gradient	Wall Function	Wedge
$ω$ $ω$	$484.269 s^{- 1}$ $484.269 s^{- 1}$	Zero Gradient	Wall Function	Wedge
$α_{t}$ $α_{t}$	$0$ $0$ (Calculated)	$0$ $0$ (Calculated)	Wall Function	Wedge
$μ_{t}$ $μ_{t}$	$0$ $0$ (Calculated)	$0$ $0$ (Calculated)	Wall Function	Wedge

Table 4.2: Boundary Conditions for Supersonic Case
Parameter	Inlet (ABC)	Outlet (GHI)	Wall (AEHIFC)	Wedges (AGHEBA + AGIFCA)
Velocity	Zero Gradient	Zero Gradient	$0.0 m s^{- 1}$ $0.0 m s^{- 1}$	Wedge
Pressure	$1.2999 N m^{- 2}$ $1.2999 N m^{- 2}$	Wave Transmissive $0.0296 N m^{- 2}$ $0.0296 N m^{- 2}$	Zero Gradient	Wedge
Temperature	$1.0388 K$ $1.0388 K$	Zero Gradient	Zero Gradient	Wedge
$k$ $k$	$3.75 \times 10^{- 5} m^{2} / s^{2}$ $3.75 \times 10^{- 5} m^{2} / s^{2}$	Zero Gradient	Wall Function	Wedge
$ω$ $ω$	$0.144 s^{- 1}$ $0.144 s^{- 1}$	Zero Gradient	Wall Function	Wedge
$α_{t}$ $α_{t}$	$0$ $0$ (Calculated)	$0$ $0$ (Calculated)	Wall Function	Wedge
$μ_{t}$ $μ_{t}$	$0$ $0$ (Calculated)	$0$ $0$ (Calculated)	Wall Function	Wedge

Results

Results for the subsonic cases are calculated from Bernoulli’s equation and the ideal gas equation as follows:

Subsonic Flow

Subsonic flow does not see a significant temperature rise. So the speed of sound remains almost constant and can be calculated as:

c = \sqrt{\frac{γ R T}{m}}

Last updated: January 29th, 2019

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Documentation

Compressible Flow: de Laval Nozzle

Overview

Geometry

Analysis type and Domain

Simulation Setup