subsonic non-steady-state gas flows in channels with inner cavities

Journal of Experimental and Theoretical Physics, Vol. 92, No. 6, 2001, pp. 991–997.Translated from Zhurnal Éksperimental’no

œ

i Teoretichesko

œ

Fiziki, Vol. 119, No. 6, 2001, pp. 1143–1150.Original Russian Text Copyright © 2001 by Breev, Rodionov, Efremov.

PLASMA, GASES

Subsonic Non-Steady-State Gas Flows in Channels with Inner Cavities

V. V. Breev, N. B. Rodionov*, and N. M. Efremov**State Scientific Center, Troitsk Institute for Innovation and Thermonuclear Studies (TRINITI),

Troitsk, Moscow oblast, 142190 Russia*e-mail: [email protected]**e-mail: [email protected]

Received February 13, 2001

Abstract—A set of gasdynamic equations is given in the general form for matter with an arbitrary equation ofstate in the case when the entropy equation is used instead of the energy equation. In the ideal gas approximationin view of viscosity, a numerical investigation is performed of non-steady-state two-dimensional flows in achannel with a cavity. The calculation results have demonstrated that, given the flow velocity and the geometryof channel and cavity, pressure pulsations arise that are due to the departure of vortices from the cavity into themain flow. The values of the amplitude and frequency of pressure pulsations are determined. If measures aretaken aimed at limiting the departure of vortices from the cavity, for example, a baffle is installed to restrict theinteraction between the main flow and gas in the cavity, one can considerably increase the flow velocity in thechannel, unaffected by the cavity. Such non-steady-state flows may be realized in MHD-generator channels,resonators of gas flow lasers, gas ducts for ventilation and gas transport systems, mufflers, whistles, etc. © 2001MAIK “Nauka/Interperiodica”.

1. INTRODUCTION

By now, a large number of experimental and predic-tion studies have been performed to investigate sub-sonic steady-state flows in smooth-walled channels ofdifferent shapes [1]. Such flows are realized in the sub-sonic part of a two-dimensional Laval nozzle, in vanechannels of turbines and compressors, in chemicalreactors for the deposition of thin films, etc. Along withthis, the study of non-steady-state flows is of certainscientific and practical interest. Here, one can singleout the problem of gas flow past a cavity in a channel.The solution of this problem will help explain the orig-ination of vortices in liquid or gas flows and study indetail the generation of the so-called vortex sound.Numerous technical devices exist in which at least onewall has a closed or open cavity, which may bring aboutthe emergence of the non-steady-state flow mode in thedevice. Such devices include MHD-generator chan-nels, resonators of gas flow lasers, whistles, gas ductsfor ventilation and gas transport systems, mufflers, andothers. In a number of cases, for example, in pulse-peri-odic gas lasers during the imposition of discharge, theflow in the resonator is non-steady-state and character-ized by the inleakage of flow into the cavity and by pos-sible emergence of acoustic disturbances in the flow.The treatment of such flows will enable one to estimatethe effect of the non-steady-state characteristics of flowor of the geometry of the flow passage, for example, onthe parameters of radiation being generated, on theelectrode processes in lasers and MHD generators, andso on. The parameters of such systems may be esti-

1063-7761/01/9206- $21.00 © 20991

mated analytically only in simple cases. The numericalcalculation of such flows helps in analyzing the dynam-ics of disturbances in gas and liquid, calculating theparameters of acoustic disturbances, and studying theeffect on these parameters of the characteristics of flowin the real geometry of the flow passage.

At present, mathematical simulation is widely usedfor numerical analysis of hydrogasdynamic problems;fairly extensive literature is available on the subject.The monograph by Godunov et al. [2] may be cited asan example. The set of equations employed includesequations of conservation of mass, momentum, andtotal energy and is closed by the equation of state. Theenergy equation may be written in several equivalentforms; however, it is usually written in terms of specificenergy or temperature. In [3, 4], the equation forentropy (which has a simpler form) was used instead ofthe energy equation, and the density ρ (or pressure P)and the entropy S were used as the independent thermo-dynamic variables. This approach is more advisablebecause, for example, under conditions of adiabaticcontractions and expansions, only ρ changes in thevariables, while in the case of (ρ, T) both variableschange (ρ, S). We will also adhere to this approach.Given in this paper is the set of equations in the vari-ables (ρ, S) and (P, S) in the general case of matter withan arbitrary equation of state. Within the ideal gasmodel in view of viscosity, numerical simulation wasperformed of subsonic gas flow in a square channelwith an inner cavity, simulating the flow passage of theresonator of a gas laser. It is the objective of this study

001 MAIK “Nauka/Interperiodica”

992

BREEV

et al

.

to perform a numerical investigation of non-steady-state flows in a channel with closed cavity, which maycause the emergence of acoustic vibrations.

2. EQUATIONS OF GAS DYNAMICS

The set of equations of gas dynamics includes thecontinuity equation

(1)

the equation of motion

(2)

and the energy equation, which may be written in sev-eral equivalent forms, with the simplest form being

(3)

Here, ρ is the density; u is the velocity vector; P is thepressure; T is the temperature; S is the entropy; and Fand Q denote the density of power and thermal effect onthe flow due to external sources and dissipation pro-cesses (viscosity and thermal conductivity), respec-tively.

The following arguments may be advanced in favorof the equation of entropy instead of energy: first, theequation for entropy is written in a simpler form, namely,only convective transfer S on space coordinates isobserved in the absence of heat sources; and, second, inprocesses with constant entropy, the set of equations (1)–(3) reduces in fact to the solution of two equations (1) and(2). The model being treated may be especially efficientwhen used in a weakly compressed liquid, i.e., whenthe velocity of gas particles is much lower than that ofsound. This enables one to study the propagation ofhydrodynamic disturbances both in gas and in liquid.This set of equations will be especially simple in thecase when the velocity of sound is constant and nosource-related terms are present.

We will use the equation defining the velocity ofsound as the equation relating the pressure and density,

(4)

The velocity of sound c and the quantities appearing inthe set of equations when this replacement is made arecalculated using the thermodynamic functions. In thegeneral case of matter with an arbitrary equation ofstate, (ρ, u, S) or (P, u, S) may be selected as indepen-dent variables for the set of equations (1)–(4). We willdiscuss these two cases in more detail. Assume that(ρ, u, S) are selected as independent variables and theinternal energy E is the preassigned function of ρ and

dρdt------ ρdiv u+ 0,=

dudt------ 1

ρ---+ —P

1ρ---F,=

dSdt------

QρT-------.=

c2 ∂P∂ρ------

S

= .

JOURNAL OF EXPERIMENTAL

S: E = E(ρ, S). Then, in view of Eq. (4), we have for thepressure gradient

In view of this, the set of equations (1)–(3) will be writ-ten as

(5)

where dϕ = dρ/ρ. In so doing, the velocity of sound, thetemperature, and the derivative of pressure with respectto entropy at constant density are determined from thethermodynamic relations

(6)

In the case of ideal gas, the expression for internalenergy has the form

(7)

where E0, S0, and ρ0 are some reference values ofenergy, entropy, and density, respectively; γ is the adia-batic exponent; CV = R/(γ – 1); and R is the gas constant.Then, we use formulas (6) to derive expressions fortemperature, velocity of sound, and the partial deriva-tive of pressure with respect to entropy,

The pressure is determined using the equation of state

The set of equations (5) is written in the variables(ϕ, u, S), which may be used for calculation in the caseof a homogeneous medium. In the case of multicompo-nent media, it is more convenient to replace ρ by P,because the pressure is continuous on the contact sur-face. We will write the set of gasdynamic equations inthe independent thermodynamic variables (P, u, S) andassume that the enthalpy h is preassigned as a functionof P and S. Then, the equation of state will be repre-sented as ρ = ρ(P, S). We differentiate this expression

—P c2—ρ=∂P∂S------

ρ

+ —S.

dϕdt------ div u+ 0,=

dudt------ c2—ϕ 1

ρ--- ∂P

∂S------

ρ—S+ +

1ρ---F,=

dSdt------

QρT-------,=

c2 ∂∂ρ------ ρ2 ∂E

∂S------

S S

= ,∂P∂S------

ρ

∂2E∂ρ∂S-------------ρ2,=

T∂E∂S------

ρ

= .

E E0ρρ0-----

γ 1– S S0–CV

-------------- ,exp=

T T0ρρ0-----

γ 1– S S0–CV

-------------- , c2exp γRT ,= =

1ρ--- ∂P

∂S------

ρ

γ 1–( )T .=

P ρRT .=

AND THEORETICAL PHYSICS Vol. 92 No. 6 2001

SUBSONIC NON-STEADY-STATE GAS FLOWS IN CHANNELS WITH INNER CAVITIES 993

with respect to time and take into account Eq. (4) defin-ing the velocity of sound to derive

In view of this, Eqs. (1)–(3) transform to

(8)

The velocity of sound, the temperature, the density, andthe derivative of density with respect to entropy at con-stant pressure are determined from the thermodynamicrelations

(9)

In the case of ideal gas, the expression for enthalpy hasthe form

where h0 is some reference value of enthalpy, and CP =Rγ/(γ – 1). Then, formulas (9) yield

The equation of state enables us to determine the den-sity as

In Eqs. (5) and (8), the equations of motion and conti-nuity without sources form a set of equations of nonlin-ear acoustics. Energy equation (3) may be solved inde-pendently if the velocity of sound is known at everyspace point or if a relation is available that relates thevelocity of sound to hydrodynamic quantities (forexample, the Poisson adiabat).

3. FORMULATION OF THE PROBLEM

We treat, within the framework of the equationsgiven above, a two-dimensional problem of gas flow ina square channel with a closed cavity in a side wall. Thesingularity of the effects observed in this case consists

dρdt------

1

c2----dP

dt-------=

∂ρ∂S------

P

+dSdt------.

1

c2----dP

dt------- ρdiv u+

∂ρ∂S------

P

QρT-------,–=

dudt------

1ρ---—P+

1ρ---F,=

dSdt------

QρR-------.=

c2 1

ρ2-----/

∂2h

∂P2---------

S

,∂ρ∂S------

P

– ρ2 ∂2h∂P∂S-------------,–= =

T∂h∂S------

P

,1ρ--- ∂h

∂ρ------

S

= = .

h h0PP0-----

γ 1–( )/γ=

S S0–CP

-------------- ,exp

T T0PP0-----

γ 1–( )/γ=

S S0–CP

-------------- , c2exp γRT ,=

∂ρ∂S------

P

ρR---.–=

ρ P/RT .=

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHY

in that, on the one hand, the cavity is an acoustic reso-nator with eigenfrequencies νi and, on the other hand,vortices may arise and be carried away by the flow ofgas or liquid past that cavity. The periodic departure ofthe formed vortices from the cavern will lead to pres-sure pulsations. The eigenfrequencies of the acousticresonator are defined by its characteristic dimensions lr

and velocity of sound c and may be represented in thegeneral form as [4]

where φ(cαi/lr) is some numerical coefficient and αi isthe damping factor for vibrations with the frequency νi.The frequency of pressure pulsations, governed by theperiodic departure of vortices arising in the cavity, isproportional to the quantity VF/lr , where F is somecoefficient depending on geometry. This system may betreated as a self-oscillatory nonlinear system [4]; i.e.,the resonator eigenfrequencies and the formation of avortex structure in the flow are interrelated. We will treatsuch a system within the ideal-gas model. For this pur-pose, we will write the set of gasdynamic equations (5) inview of Eqs. (6) and (7) in the form

(10)

where Q is the heat released by the work of viscousforces, µ is the viscosity coefficient, and λ is the ther-mal conductivity coefficient for gas. We will restrictourselves to the case in which the gas flow velocity ismuch lower than the velocity of sound c0. In this case,one can ignore the heat released under conditions ofviscous dissipation. We will likewise ignore the thermalconductivity. The term related to viscosity is determin-ing in a very thin layer. For the characteristic geometricdimensions and gas dynamic parameters treated by us,the thickness of laminar boundary layer was δ ≈ 0.003 m[5]. In view of these assumptions, the set of equations(10) will be written as

(11)

ν iclr

---φ clr

---α i ,=

dϕdt------ div u+ 0,=

dudt------ c2—ϕ+

µρ---∆u=

13---µ

ρ---— div u( ),+

dSdt------

λρ---∆T

T-------=

QρT-------,+

dϕdτ------

∂V x

∂x---------

∂Vy

∂y---------

βV x

x---------+ + + 0,=

dV x

dτ--------- ∂ϕ

∂x------+

νc0----

∂2V x

∂x2-----------

∂2V x

∂y2-----------+

=

+ν

3c0--------

∂2V x

∂x2-----------

∂2Vy

∂y∂x------------+

,

SICS Vol. 92 No. 6 2001

994 BREEV et al.

where β = 0, 1 for plane and cylindrical geometry,respectively; Vx and Vy are the components of the veloc-ity vector u along the x and y axes, respectively, nor-malized to the sound velocity c0; and ν= η/ρ is the kine-matic viscosity. The variable τ = c0t is introducedinstead of time. The set of equations (11) is used to per-form a numerical analysis of the dynamics of nonsta-tionary interaction of gas flow in a square channel witha cavity. We will write Eqs. (11) in a compact matrixform,

(12)

where

The initial conditions are preassigned in the form

The form of the boundary conditions depends on theconcrete formulation of the problem.

4. NUMERICAL MODEL

The total approximation technique [9] is used to findthe numerical solution of Eq. (12). Two problems aresolved successively at the step ∆τ,

(13)

(14)

dVy

dτ--------- ∂ϕ

∂y------+

νc0----

∂2Vy

∂x2-----------

∂2Vy

∂y2-----------+

=

+ν

3c0--------

∂2Vy

∂y2-----------

∂2V x

∂x∂y------------+

,∂S∂t------ 0,=

dΦdτ------- AΦ ψ+ + 0,=

ΦϕV x

Vy

, A

0∂∂x------ ∂

∂y-----

∂∂x------ 0 0

∂∂y----- 0 0

,= =

Ψ

βV x

x---------

νc0----∆V x

ν3c0--------∂div V

∂x-----------------+

νc0----∆Vy

ν3c0--------∂div V

∂y-----------------+

.=

Φ x y 0, ,( ) Φ0 x y,( ).=

∂Φx

∂τ--------- Ax

∂Φx

∂x--------- Ψx+ + 0,=

∂Φy

∂τ--------- Ay

∂Φy

∂y--------- Ψy+ + 0,=


where

For Eqs. (13) and (14), use is made of the explicit two-step difference scheme of [6–8] modified for this prob-lem,

where

ΦxϕV x

= , ΦyϕVy

,=

AxV x 1

1 V x

= , AyVy 1

1 Vy

,=

Ψx

βV x

x---------

νc0---- ∆V x

∂divV3∂x

---------------+

,=

Ψy

0

νc0---- ∆Vy

∂divV3∂y

---------------+ .=

Φ̃xk 1±τ ∆τ+ Φxk 1±

τ= ± Bxk 1±τ Φxk 1±

τ Φxkτ–

∆x-----------------------------∆τ Ψxk

τ ∆τ ,+

Φ̃yk 1±τ ∆τ+ Φyk 1±

τ= ± Byk 1±τ( )

Φyk 1±τ Φyk

τ–∆x

-----------------------------∆τ Ψykτ ∆τ ,+

Bxk 1±τ

V xk 1± V xk+2

---------------------------- 1

1V xk 1± V xk+

2----------------------------

,=

Byk 1±τ

Vyk 1± Vyk+2

---------------------------- 1

1Vyk 1± Vyk+

2----------------------------

,=

Ψxkτ

βV x

x---------

ν ∆V x∂div V

3∂x----------------+

k

,=

Ψykτ

0

ν ∆Vy∂div V

3∂y----------------+

k

,=



The expressions for and are similar to

those for and , but are constructed with due

regard for the form of and . The foregoing scheme is of the second order of

accuracy with respect to the space coordinates and ofthe first order with respect to time [9]. The conditionsof stability of the scheme are valid with a margin.

We will calculate the boundary conditions usingEqs. (1) and (2) written in the characteristic form,

(15)

For visualizing the velocity field, we will calculatethe quantities

and construct vector diagrams of the vector field ( , ).By way of a test, a calculation was performed of

one-dimensional flat, cylindrical, and spherical soundpulses on the condition that the velocity of sound is muchhigher than the rate of displacement of gas particles. Suchcalculations were performed in accordance with the pro-cedure described in [6]. The results obtained (gasdynamicparameters of pulses) were compared with analyticalexpressions [10], and the laws of conservation in thiscase were checked for validity. In order to check thecomputer codes, the flow rate in the inlet and outletcross sections was monitored. The calculation resultsdemonstrated that, at τ ≥ 20, the values of flow rateagreed within 1%. The viscous steady-state plane-par-allel flow in the channel was calculated as well, and aparabolic velocity profile was obtained to a good accu-racy [12].

5. NUMERICAL CALCULATION RESULTSWithin the problem set, we will treat the flow of air

in a square channel with a cavity in a side wall, whichsimulates the flow passage of a gas laser. A schematicdiagram of such a device is given in Fig. 1. The follow-ing geometric dimensions and parameters of flow and

Φ̃˜ xk

τ ∆τ+Φxk

τ= + Axkτ ∆τ+ Φ̃xk 1+

τ ∆τ+ Φ̃xk 1–τ ∆τ+

–2∆x

-----------------------------------∆τ Ψ̃xkτ ∆τ+ ∆τ ,+

Φ̃˜ yk

τ ∆τ+Φyk

τ= + Aykτ ∆τ+ Φ̃yk 1+

τ ∆τ+ Φ̃yk 1–τ ∆τ+

–2∆x

-----------------------------------∆τ Ψ̃ykτ ∆τ+ ∆τ ,+

Axkτ ∆τ+ V xk 1

1 V xk

,= Aykτ ∆τ+ Vyk 1

1 Vyk

.=

Ψ̃xkτ ∆τ+ Ψ̃yk

τ ∆τ+

Ψxkτ Ψyk

τ

Φ̃xk 1±τ ∆τ+ Φ̃yk 1±

τ ∆τ+

dϕdτ------

dV x

dτ---------±

V xβx

---------+ 0,dxdτ------ V x 1,±= =

dϕdτ------

dVy

dτ---------±

Vyβy

---------+ 0,dydτ------ Vy 1.±= =

V xV x

V x2 Vy

2+----------------------,= Vy

Vy

V x2 Vy

2+----------------------=

V x Vy


cavity were selected for calculation: the channel lengthalong the flow L = 0.3 m, the channel width H = 0.1 m,the cavity length lc = 0.13 m, the cavity height hc =0.04 m, and the length and height of the throat connect-ing the channel and the resonator cavity lth = 0.1 m andhth = 0.03 m. The pressure Pout at the channel outlet wastaken to be 0.1 atm with a temperature of 300 K. Theinlet velocity was varied from 10 to 100 m/s. The cal-culations were performed in a square mesh grid. Thestep in space was assumed to be ∆h = 0.002 m, and thestep in time ∆τ = CC∆h, where CC is the Courant num-ber. It was assumed that CC = 0.9 [8, 9]. The calculationwas performed during a period of time τ ≈ 0–0.02 s. Inthe inlet cross section, the velocity was preassigned,and in the outlet cross section, the pressure. With t = 0in the calculation region, zero velocity was preas-signed, and the pressure was assumed to be equal to theoutlet pressure. The flow relaxed to a steady state inperiods of time of about L/V. The transition processcaused pressure pulsations which either decayed or didnot decay. It has been assumed that, if the pulsationamplitude continues to decrease over periods of timeτ @ L/V, such a vibrational process is decaying, andconversely, if the amplitude did not change with time,pressure pulsations occur in the flow. Also performedfor monitoring were calculations during a period of timeconsiderably exceeding L/V (≈0.6 s), whose results dem-onstrated the presence or absence of pressure pulsa-tions, depending on the selected parameters.

The calculation results demonstrate that, in theentire range of the treated parameters of the problem,the pressure oscillation with a frequency ν ≈ 800 Hzarises during the initial period in the cavity over itsentire depth, with the amplitude of this oscillationreaching a value of 8% pressure at the channel outlet.

lth

hc

hth

lc

L

AB

C

y

xH

1

2

u

Fig. 1. The channel and cavity scheme: L and H, the channellength and width, respectively; lc and hc, the length andwidth of the resonator cavity, respectively; lth and hth, thelength and width of the throat connecting the channel andresonator cavity, respectively. The broken line indicates twobaffles (1, 2). The circles indicate the points A, B, and C,where the pressure pulsations are calculated.

SICS Vol. 92 No. 6 2001

996 BREEV et al.

In the course of time, depending on the velocity of flowin the channel, this oscillation may either decay (at V ≤40 m/s) or its amplitude reaches some steady-statevalue, and this oscillation is imparted to the main flow(V ≥ 60 m/s). Figures 2 and 3 give fragments of the pat-tern of velocity field ( , ). An analysis of thevelocity field in the channel and cavity reveals that avortex forms in the cavity. With a main flow velocity ofup to 40 m/s, this vortex is in the cavity and its center isstationary (Fig. 2). When the velocity increases from 40to 60 m/s, the vortex moves in the cavity; simulta-neously, much smaller vortices are formed in the cavity.The vortices, which form in the vicinity of the flow, arecarried periodically one by one into the main flow;therefore, as the velocity increases, the decay of pres-sure pulsations becomes weaker. When the inlet veloc-ity reaches a value of about 60 m/s, all of the vorticesformed in the cavity depart periodically one by one into

V x Vy

Fig. 2. A fragment of the pattern of velocity field ( , )in the cavity and channel with the inlet velocity of flow of20 m/s.

Vx Vy

–0.08

–0.04

0

0.04

0.08

δP/P

out

0 0.002 0.004 0.006 0.008 0.010t, s

Fig. 4. The pressure pulsations (δP/Pout) as a function oftime, with the inlet velocity of 60 m/s, at the points A, B, andC (Fig. 1). Dot-and-dash curve, at point A; solid curve, atpoint B; dashed curve, at point C.


the main flow. In so doing, the secondary vortex thatforms at the left-hand wall of the throat is comparablein magnitude to the initial vortex (Fig. 3). From thismoment on, the pressure oscillation ceases to decay.The flow in the channel starts “making noise.” The cal-culation results demonstrate that the period of repeti-tion of the velocity field pattern corresponds to a fre-quency of approximately 800 Hz. One can assume thatthe periodic departure of vortices from the cavity main-tains the pressure oscillation. Figure 4 gives the pat-terns of pressure distribution as a function of time at thepoints A, B, and C (Fig. 1). One can see that the oscil-lation occurs with a frequency of ≈800 Hz. The calcu-lations performed with a finer mesh grid produced thesame results.

We performed calculations of flow as a function ofthe geometric dimensions of the cavity. It has beenfound that an increase in the cavity height hc by approx-

Fig. 3. A fragment of the pattern of velocity field ( , )in the cavity and channel with the inlet velocity of flow of60 m/s.

Vx Vy

Fig. 5. A fragment of the pattern of velocity field ( , )in the cavity and channel in the presence of baffles on theside of undisturbed flow. The inlet velocity of flow, 60 m/s.

Vx Vy



imately 20% causes very little change in the frequencybut leads to a decrease in the value of critical velocityat which the pressure pulsations cease to decay. In ourcase, this critical velocity is 20 m/s. Analogously, a20% increase in the width of the resonator cavity leadsto a decrease in the critical velocity by half.

The calculation results have demonstrated that onecan substantially reduce the pressure pulsations if a thinbaffle 1 1 cm in size or less is installed between the cav-ity and channel on the side of undisturbed flow at thebeginning of the cavity (see Fig. 1). As a result, the baf-fle stabilizes the initial flow in the cavity and preventsthe origination of vortices in the vicinity of the mainflow in the cavity. Therefore, the velocity at which thepressure oscillation decays increases to 100 m/s (nocalculations were performed at higher values of veloc-ity). The pattern of velocity field in this case is similarto that given in Fig. 2. The provision of yet another baf-fle (shown at 2 in Fig. 1) at the end of the cavity nolonger affects the behavior of flow in the channel; how-ever, two conjugate vortices are formed in the cavity(Fig. 5).

Note that, because the cavity dimensions are notsmall compared with the region under investigation, itwould be more correct to treat the entire region as acomplex acoustic resonator, when the channel lengthalong the flow L and the channel width H affect the crit-ical velocity and the frequency of pressure oscillation.Therefore, the obtained results are qualitative and, for aconcrete device, calculations must be made with dueregard for concrete geometry.

This procedure is also suitable for investigating anon-steady-state flow of liquid in a channel with a cav-ity, if the equation of state for ideal gas is replaced bythe respective equation for liquid in the form of Tate’sformula [11].

6. CONCLUSIONIn this paper, a set of gasdynamic equations is given

in the general form with an arbitrary equation of statefor the case when the entropy equation is used insteadof the energy equation. In the ideal gas approximationin view of viscosity, a numerical investigation is per-formed of a non-steady-state two-dimensional subsonicflow in a channel with a cavity, simulating the flow pas-sage of the resonator of a gas laser. It has been foundthat, in such cases, given some velocity of incidentflow, pressure pulsations may arise in this flow that aredefined by the characteristics of undisturbed flow and


by the geometry of the flow passage. The frequency andamplitude of these pulsations have been analyzed. Ithas been found that a 20% variation of the geometricdimensions of the cavity causes the critical velocity, atwhich acoustic disturbances occur, to vary by a factorof two. The calculation results have demonstrated thatthe pressure oscillation is associated with the departureof vortices arising in the cavity and that the frequencyof this oscillation coincides with the frequency ofdeparture of vortices from the cavity. The provision ofbaffles in the closed cavity in the vicinity of the flowmay considerably reduce the effect of the cavity on themain flow and increase the critical velocity of flow atwhich pressure pulsations arise in the channel.

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11. R. Courant and K. O. Friedrichs, Supersonic Flow andShock Waves (Interscience, New York, 1948; Inostran-naya Literatura, Moscow, 1960).

12. H. Schlichting, Boundary Layer Theory (McGraw-Hill,New York, 1955; Inostrannaya Literatura, Moscow,1956).

Translated by H. Bronstein

SICS Vol. 92 No. 6 2001

subsonic non-steady-state gas flows in channels with inner cavities

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