CFD simulation in Laval nozzleSIAE 090441313AbstractWe aim to simulate the quasi one dimension flow in the Laval nozzle based on CFD computation in this paper .We consider the change of the temperature ,the pressure ,the density and the speed of the flow to study the flow.The analytic solution of the flow in the Laval nozzle is provided when the input velocity is supersonic.We use the Mac-Cormack Explicit Difference Scheme to slove the question.Key words :Laval nozzle ,CFD,throat narrow.ContentsAbstract .................................................. . (1)Introduction .............................................. .. (2)Simulation of one-dimensional steady flow (3)Basisequations ................................................. (3)Dimensionless .......................................... . (10)Mac -Cormack Explicit Difference Scheme (11)Boundaryconditions ................................................ (13)Reference .............................................. (13)Annex .................................................. .. (14)IntroductionLaval nozzle is the most commonly used components of rocket engines and aero-engine, constituted by two tapered tube, one shrink tube, another expansion tube.Laval nozzle is an important part of the thrust chamber. The first half of the nozzle from large to small contraction to a narrow throat to the middle. Narrow throat and then expandoutwards from small to big to the end. The gas in the rocket body by the front half of the high pressure into the nozzle, through the narrow throat to escape by the rear half. This architecture allows the speed of the air flow changes due to changes in the jet cross-sectional area, the airflow from subsonic to the speed of sound, until accelerated to transonic. So, people flared nozzle called transonic nozzle. Since it was invented by the Swedish Laval, also known as Laval nozzle. Analysis of the principle of the Laval nozzle. The rocket engines of the gas flow in the combustion chamber under pressure, after the backward movement of the nozzle into the nozzle . At this stage, the gas movement follow the principle of "the fluid moves in the tube , the small cross-section at the flow rate large sectional large flow velocity", thus accelerating airflow.Laval nozzleWhen you reach the narrow throat, the flow rate has exceededthe speed of sound. Transonic fluid movement they no longer follow the principle of "cross-section at small velocity, at a flow rate of small cross-section large", but on the contrary the larger cross-sectional flow faster. The gas flow speed is further accelerated to 2-3 km / sec,equivalent to 7-8 times the speed of sound, thus creating a great thrust. The Laval nozzle fact played the role of a "flow rate Enlargement Device". In fact, not just rocket engines, missile nozzle is this horn shape, so the Laval nozzle weapons has a very wide range of applications.Simulation of one-dimensional steady flow1.Basis equationsAs we know,Laval nozzle is a zooming nozzle flow channel to narrow further expansion.Allows the airflow to further accelerate to reach the speed of sound at the throat into a supersonic flow.Now,we want to simulate the quasione-dimension flowing.Firstly,we will analysis on theory.The flow is isentropic,so we can apply the following equations.(1)Continuity equation:In the flow, we need to consider the following physical quantities.The pression ,the temperature ,the speed of the fluid and the cross-section .They are respectively represented by P,T,u,A. We apply the conservation of the mass.we will obtain this equation.))()((du u dA A d uA +++=ρρρAnd then we get=++ρρd u du A dA(2)Equation of momentum(in the direction of the axis) According to the theory of momentum:dAdP P dA A dP P PA uAu du u uA )2())(()(++++-=-+ρρThe simplification of this equation isdP udu -=ρ(3)Energy equation)2(2=+=+=udu dh v h d dh tIdeal gas equation of stateRTMP ρ=R is ideal gas constant,R=8.314J/g/K. M is the masse per mole.(4)The equation of ThermodynamicsP dPRT dT C dP T V T dT C dS dT C dh VdP dh TdS P e h PdV de dS p p p -=-=⇒=-=+=+=;,;V ,T Because the flow is isentropic,sodS=0And we use the equation of momentum,we have1)(T P -∆=∆=∆γγT RC p）（Combine with others equations,we result withRTγ=2uWe called u the speed of sound,we noted a.RTγ=2aWe apply the continuity equation)1(A dA 22-=a uWe defined the Mach numbera u =MIf we have the relation as)48.0tanh(347.0398.1A -+=xWe have the figure ○1Sou du M A A )1(d 2-= M>1,supersonic If dA<0,we have du>0. If dA>0,we have du<0. M<1,subsonicIf dA<0,we have du>0. If dA>0,we have du<0.This is the reason why this architecture allows the speed of the air flow changes due to changes in the jet cross-sectional area, the airflow from subsonic to the speed of sound, until accelerated to transonic.We have the consequence as followsPAMRTMAMP T R t tγγγγγ=-+⋅=-+)1(212)2111(m&1)211(;)211(T 22--+=-+=γγγγM P P M T ttThen we replace P and T in this equation.The consequence will become122)211(211m--+⋅-+=γγγγγM AMP T M Rt t&To simplify)1(212)2111(m-+-+=γγγγMAM T P R tt &In this equation,the variable is the much number,as the speed of the flow is from subsonic to supersonic ,so we can suppose that there exist a critical section where M equal to 1.Then)1(212)21211(1-+*+-+=γγγγMM AAFigure ○2This section is called narrow throat.The same method,we can obtain121122)21121()21121(21121-*-**-++=-++=-++=γγγγγρργγγγM M P PM T TFigure ○3We know the section in narrow throat is minimum.])2111)()1())1(2)1(()111[()(1)1(212)1(212+-+-+++-⋅-+-+-+=γγγγγγγγγγMM M M K AmdM d &we can judge that the function attains the maximum or not)1(212)2111()(f -+-+=γγγMM M2 DimensionlessCombining CFD with one-dimension flow theory,we make the variables dimensionless.According to the condition initial which is given .We note0'0''0'0'0'u lt t A A A lx x u u u T T T ======ρρρWe use the variable dimensionless to represent theequations.And the equations have the following changes (1)Continuity equation'''''''''''''''''''''''ln t 0t x v x A v x v xv x A A v x v ∂∂-∂∂-∂∂-=∂∂⇒=∂∂+∂∂+∂∂+∂∂ρρρρρρρρ‘‘ (2)Equation of momentum)(1v '''''''''''x T x T x v v t ∂∂+∂∂-∂∂-=∂∂ρργ (3)Energy equation)ln ()1(T '''''''''''x A v x v T x T v t ∂∂+∂∂--∂∂-=∂∂γ3.Mac-Cormack Explicit Difference SchemeThen we use the Mac-Cormack Explicit Difference Scheme,theprincipal of this theory is using the surrounding points to present differential parts of a point and we consider the question with one dimension.The distance between two points is h.0222003022200103013003320022000)(2)()(2)()()(!31)()(!21))((f f(x)x f h x f h f f x f h x f h f f hx x hx x x x xf x x x f x f x x ∂∂+∂∂-=∂∂+∂∂+=-=+=-∂∂+-∂∂+∂∂-+≈So we can use two points adjacent to present the differential parts.20310223102)(2)(h f f f x f hf f x f -+=∂∂-=∂∂Using this method,we make an estimation and correct the error. Estimationx nA A v x v v x v t i i t i t i t i t i t i t i t i t i t i ∆-⋅-∆-⋅-∆--=∂∂+++111ln )(ρρρρρ correct the errorx v t t t i t t i t t i t t i ∆--=∂∂∆+-∆+∆+∆+1)(ρρρIntermediate valuet t t i t i t t i ∆∂∂+=∆+)(ρρρ Then the equation has the following change))()((21)(t ti ti av t t t ∆+∂∂+∂∂=∂∂ρρρAt the moment t ,we will know the value in the whole plan . And we define).....,,,,min(54321t N t t t t t ii ii t t t t t t t a v x c t ∆∆∆∆∆∆=∆+∆=∆4.Boundary conditionsHyperbola equation has two characteristics lines.When one of the characteristics lines enter the flow zone .We admit a parameter to be fixed ,otherwise when one of the characteristics go out the flow zone ,we admit a parameter to be a variable depends the time.Applying this theory ,we can determine the boundary conditions.Reference ：[1]章利特，高铁瑜，夏庆锋，徐廷相.拉瓦尔喷管的准一维定常流动.中国科技论文在线。
