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Numerical simulations have been carried out for a supersonic three-dimensional rectangular arc nozzle, where a secondary flow toward the center of the curvature occurs due to the shape of the nozzle. It is known that secondary flow causes longitudinal vortices to form near the wall of the nozzle corner, making the nozzle outlet flow unstable and induces loss of transport energy. When the working fluid is a condensable gas with relatively large latent heat such as moist air or steam, rapid accelerated expansion in the nozzle causes non-equilibrium condensation due to supersaturation. After the release of latent heat during phase transition, nozzle flow continues expanding at an equilibrium saturation condition. In the absence of foreign particles, e.g. ions or dust particles, condensation nuclei are formed in the gas itself causing non-equilibrium homogeneous condensation. Supersonic nozzle flow properties vary considerably due to the occurrence of condensation phenomenon. The objective of this study is to investigate the effect of non-equilibrium homogeneous condensation on the longitudinal vortices which form in the range close to the corner of rectangular arc nozzle numerically.

The instability of a boundary layer over the concave surface results from the imbalance between the centrifugal force and the wall-normal pressure gradient in case of supersonic rectangular nozzle flow [

There has not been much study about the interference between non-equilibrium homogeneous condensation and the corner flow vortices. In general, air as a working fluid contains humidity which induces non-equilibrium homogeneous condensation due to adiabatic expansion. Therefore, it is important to understand the effect of non-equilibrium homogeneous condensation on the corner flow vortices in order to delay the laminar-turbulent flow transition and to reduce the total pressure loss.

In case of supersonic rectangular nozzle flow, high-speed expansion of moist air causes the formation of condensate particles [

This research aims to investigate the effect of non-equilibrium homogeneous condensation on the corner flow of a three-dimensional supersonic rectangular arc nozzle numerically. Density distributions along with the flow vorticity near the nozzle corner have been analyzed to know the influence of non-equilibrium homogeneous condensation on the flow vortices in cases of dry air and moist air.

In this research, numerical calculations were performed using the unsteady three-dimensional compressible Navier-Stokes equations in combination with the equations of continuity, energy, turbulent kinetic energy, specific dissipation rate, conservation of mass of the liquid phase, and conservation of the number density of droplets with homogeneous nucleation. Discretization was performed spatially by the cell-centered type finite volume method. Furthermore, a third-order accurate MUSCL TVD (Total Variation Diminishing) schemes based on Roe’s approximate Riemann solver [

∂ Q ∂ t + ∂ E ∂ x + ∂ F ∂ y + ∂ G ∂ z = ∂ E v ∂ x + ∂ F v ∂ y + ∂ G v ∂ z + I + S (1)

where Q is the conservative vector; E, F and G are inviscid flux vectors; E_{v}, F_{v}, G_{v} are viscosity flux vectors. I and S are the source terms corresponding to turbulence and condensation, respectively. The governing equation systems that are non-dimensionalized with reference values at the inlet conditions upstream of the nozzle are mapped from the physical plane into a computational plane of a general transform.

Q = [ ρ m ρ m u ρ m v ρ m w E s ρ m k t ρ m k l ρ m ω ρ m g ρ m n ] , E = [ ρ m u ρ m u 2 + p ρ m u v ρ m u w ( E s + p ) u ρ m k t u ρ m k l u ρ m ω u ρ m g u ρ m n u ] , F = [ ρ m v ρ m u v ρ m v 2 + p ρ m v w ( E s + p ) v ρ m k t v ρ m k l v ρ m ω v ρ m v g ρ m v n ] , G = [ ρ m w ρ m u w ρ m v w ρ m w 2 + p ( E s + p ) w ρ m k t w ρ m k l w ρ m ω w ρ m w g ρ m w n ] (2)

Numerical modelling of non-equilibrium homogeneous condensation phenomenon consists of two equations for condensate mass fraction g and the number of droplets per unit mass n [

∂ ( ρ m g ) ∂ t + ∂ ( ρ m g u ) ∂ x + ∂ ( ρ m g v ) ∂ y + ∂ ( ρ m g w ) ∂ z = 4 π 3 ρ l ( r c 3 I F + 3 ρ m n r 2 d r d t ) (3)

∂ ( ρ m n ) ∂ t + ∂ ( ρ m n u ) ∂ t + ∂ ( ρ m n v ) ∂ t + ∂ ( ρ m n w ) ∂ t = I F (4)

Reference [_{t}-k_{l}-ω turbulence model [_{t0}, laminar kinetic energy k_{l0} and specific dissipation rate ω_{0} in case of rectangular arc nozzle flow. The y^{+}-u^{+} distribution of the flow in case of flat plate was consistent with the experimental value [

Initial calculation conditions were fixed at the nozzle inlet while the back pressure was not fixed at the nozzle outlet. Initial stagnation pressure p_{0}, temperature T_{0}, and initial degree of relative humidity ϕ 0 were set at 102 kPa, 287 K and 60%, respectively, in case of moist air. In addition, initial stagnation pressure p_{0}, temperature T_{0}, and initial relative humidity ϕ 0 were set at 102 kPa,

287 K and 0%, respectively, in case of dry air. In both cases, initial turbulent kinetic energy k_{t0}, laminar kinetic energy k_{l0} and specific dissipation rate ω_{0} were set at 0.124490 J/kg, 0 J/kg, and 364.955 1/s, respectively.

Simulations of the rectangular supersonic arc nozzle flow in cases of dry air and moist air have been carried out until the residual error has reached the order of 10^{−}^{6}. In

As shown in

Distributions of condensate mass fraction g, nucleation rate I_{F}, and static pressure p/p_{0} are illustrated for the non-equilibrium homogeneous condensation phenomenon of moist air in

Besides, the end of non-equilibrium condensation zone at x/H = 1.4 is indicated by the maximum point of the static pressure rise and after this point the flow again becomes isentropic.

In case of dry air, _{dry}, where the absolute value of density gradient (dρ/dy) is minimum and δ_{dry.max} is the maximum thickness of δ_{dry}. Extension of vorticity variation region toward the lower wall at the nozzle corner causes the position of δ_{dry} to move away from the upper wall as the measurement position approaches the sidewall. Therefore, it can be said that the change in density has a strong correlation with the change in vorticity. Further, δ_{dry.max} becomes larger at x/H = 2 compared with that of x/H = 0.

_{dry.exp}. Comparison between the calculated δ_{dry.max} and measured δ_{dry.exp} [

that the value of δ_{dry.max} is almost the same as the δ_{dry.exp} of schlieren photograph at each position. These results confirm the relation between the experimental datum and present simulation outputs.

Four cross-sectional planes (x/H = 0, 1, 1.4, and 2) in _{moist} i.e., minimum absolute value of density gradient (dρ/dy) departs further from the upper wall as the measuring position reaches the sidewall. This is attributed to the extension of vorticity variation region toward the lower wall at the corner of the nozzle. Further, δ_{moist.max} which indicates the maximum value of δ_{moist} becomes larger at x/H = 2 compared with that of x/H = 0.

In _{dry.max} and δ_{moist.max} in cases of dry air and moist air are indicated at x/H = 0, 1, 1.4 and 2. From this table, it can be seen that the non-equilibrium homogeneous condensation that occurred downstream of the nozzle throat reduces the thickness compared with the δ_{dry.max} at x/H = 1 and 1.4. Therefore, it can be deduced that the latent heat released by the non-equilibrium homogeneous condensation suppresses the development of longitudinal vortices near the nozzle corner.

Measurement positions | Thickness for dry air δ dry . max [mm] | Thickness for moist air δ moist . max [mm] | δ dry . max − δ moist . max δ dry . max × 100 [ % ] |
---|---|---|---|

x/H = 0 | 0.8568 | 0.8568 | 0 |

x/H = 1 | 0.9996 | 0.952 | 4.76 |

x/H = 1.4 | 1.1186 | 0.9044 | 19.15 |

x/H = 2 | 1.19 | 1.0472 | 12.0 |

This numerical study has been set out to investigate the effect of non-equilibrium homogeneous condensation on the corner flow of a supersonic rectangular arc nozzle. The results obtained are as follows:

1) An increase of static pressure downstream of the nozzle throat in case of moist air indicated the non-equilibrium homogeneous condensation. Simulated results of static pressure distribution were in good agreement with the experimental results.

2) Change in density at the nozzle corner flow had a strong correlation with the change in vorticity. This is because the extension of vorticity variation region toward the lower wall caused the position of density gradient with minimum absolute value to move further away from the upper wall.

3) In the range of the nozzle corner, surface of condensation onset had a complicated structure due to an interaction between the boundary layer and longitudinal vortex.

4) Latent heat released by the non-equilibrium homogeneous condensation suppressed the development of longitudinal vortices near the nozzle corner which resulted in the reduced thickness of δ_{moist.max}.

The authors declare no conflicts of interest regarding the publication of this paper.

Haque, S., Matsuo, S. and Kim, H.-D. (2021) Three-Dimensional Structure of Non-Equilibrium Homogeneous Condensation Flow in a Supersonic Rectangular Nozzle. Open Journal of Fluid Dynamics, 11, 55-66. https://doi.org/10.4236/ojfd.2021.112004

p Static pressure

Q Conservative vector

E, F, G Inviscid flux vector

E_{v}, F_{v}, G_{v}_{ }Viscosity flux vector

I Turbulence vector

S Condensation vector

T Temperature

R Radius of curvature

ρ Density

u Velocity component in x-axis direction

v Velocity component in y-axis direction

w Velocity component in z-axis direction

n Number of condensate particles

g Condensate mass fraction

k_{t} Turbulent kinetic energy

k_{l} Laminar kinetic energy

ω Specific dissipation rate

ϕ Relative humidity

H Nozzle throat height

σ Surface tension

r_{c} Critical cluster radius

I_{F} Nucleation rate

r Droplet average radius

Subscripts

0 Stagnation point

m Mixture of air and vapor

l Liquid