International Journal of Theoretical & Applied Sciences, 9(1): 35-42(2017)
ISSN No. (Print): 0975-1718 ISSN No. (Online): 2249-3247 Potential Equations and Pressure Coefficient for Compressible Flow: Comparison between Compressible and Incompressible Flow in Aerodynamics Menka Yadav * and Santosh Kumar Yadav ** *Research Scholar, J.J.T. University. Rajasthan, India ** Director (A&R), J.J.T. University, Rajasthan, India (Corresponding author: (Corresponding author: Menka Yadav) (Received 02 March, 2017 accepted 05 April, 2017) (Published by Research Trend, Website: www.researchtrend.net) ABSTRACT: We derive the potential equation for slender bodies and seek to understand the flow field equations for subsonic, supersonic and transonic flow in framework of small perturbation. Large amount of heat and mass can be transferred in a efficient way between the surface and fluid when flow is released against the surface. When aircraft passes through several distinct regions, the flow develops a velocity and pressure profile. In stagnation-region large scale turbulent flow affects transfer coefficient. At the face of object total pressure is higher than behind the object. Profile slopes shows that compressible and incompressible flows are related via certain equations. Zero Mach number incompressible medium causes pressure disturbances to move uniformly in all directions. Flow of heat and mass transfer is strongly affected by the geometry of the device. Keywords: Mach number, Pressure drag, Shock wave, Slender bodies, Velocity profile. I. INTRODUCTION called lift, which acts on the wing. Velocity varies along the wing chord and in the direction normal to its surface. Flow having significant changes in fluid density are The region adjacent to the wall, where the velocity known as compressible flow or flow with Mach number increases from zero to freestream value is known as the greater than 0.3 is treated as compressible. Compressible boundary layer. Viscous forces are predominant inside flow can be classified into subsonic, supersonic and the boundary layer. The static pressure outside the hypersonic, based on the flow Mach number. This boundary layer acting in the direction normal to the behaviour is mostly displayed by gases. In many field s surface is trnasmitted to the boundary. Pressure like jet engines, high speed aircraft, rocket motors, high coefficient becomes zero across boundary layer. If speed entry into a planetary atmosphere, gas pipelines interlayer friction is neglected between the streamlines, etc., study of compressible flow is relevant. In testing of outside the boundary layer, then it can be treated as supersonic flows, supersonic wind tunnels are used. invicid. Invicid flow is also called potential flow. In compressible flow, significant changes in velocity Pressure can be invariant across the boundary layer and and pressure results in density variations throughout the can be obtained in the field outside the boundary layer. flow field. Due to large temperature variations, density Furthermore pressure in the freestream is impressed variations takes place. But in incompressible flow, through the boundary layer. Fig. 1. material density remains constant. We make use of equations in flow field and tackle the problem by making some simplifications to the equation, depending on the type of flow to which it is to be applied. For certain flows, the equation can be reduced to an ordinary differential equation of a simple linear type. It can be reduced to nonlinear ordinary differential equation for some other type of flows. High pressure is created by the air flow over wing's bottom surface as compared to the top surface. Due to it, there is a net resultant force Fig. 1. component, normal to the freestream. Flow direction, Yadav and Yadav 36
Airfoil generates lift due to imbalance of pressure V= V + u distribution over the top and bottom surface of airfoil. x ∞ V= v …(5) Curved surface helps to reduce drag. In aerodynamics, y static pressure is due to random motion of gas molecules V= w z and pressure we would feel if moving along with the flow, and stagnation (total) pressure exist if flow were Small disturbances are introduced by slender or planar slowed down isentropically to zero velocity. bodies. In this case perturbation velocities are less than Combination of static and total pressure allows us to main velocity components. Then: measure speed at a given point. Vx ≈ V ∞
D = D + D Vy << V ∞ Total drag due to Drag due to Drag due to Vz << w ∞ separation Viscous effects skin friction For small angle of attack, Called profile drag Vx = V + u ∞ …(6) Vy = V
Less for laminar More for laminar From equation (3) and (5), we have More for turbulent less for turbulent (1−M 2 )φ ++= φ φ 0 …(7) xx yy zz II. POTENTIAL EQUATION FOR SUBSONIC , Where M is Mach number and φ is perturbation SUPERSONIC AND TRANSONIC FLOW FOR SLENDER potention and higher order terms are neglected. Using BODIES small perturbation theory and binomial theorem, relation between the local Mach number M and free stream Continuity equation for a steady, invicid 3-D flow is:- Mach number M∞ can be expressed as, ∇(ρ V ) = 0 (1) u γ −1 M2 =1 + 2 1 + M2 M 2 …(8) The above equation can be written as V 2 ∞ ∞ ∞ V ∂VVx∂ y ∂ z ∂ρd ρ ∂ ρ Combining equation (7) & (8), ρ +++ Vx + V y + V z = 0 (2) ∂∂∂xyz ∂ x ∂ y ∂ z 22 2γ − 1 2 (1−M∞ )φφφxx ++= yy zz M ∞ φφ xx 1 + M ∞ …(9) For isentropic process V∞ 2 ρ= ρ (P ) The above non-linear equation is valid for subsonic, transonic and supersonic flow. Equation (9) can be Velocity components in terms of φ can be written as:- written as
2 φ2 φ φ 2 φ φ M 2 u γ −1 1−x φ +− 1y φ +− 1z φ − 2 x y φ (1−++=M2 )φφφ 2∞ 1 + M2 (1 − M 2 ) φ 2xx 2 yy 2 zz 2 xy ∞ xx yy zz 2 ∞ ∞ mm a a a a 1− M ∞ V∞ 2 …(10) φ φ y z φz φ x 2 +φyz + φ zx = 0 (3) M u a2 a 2 If ∞ << 1 …(11) 2 1− M∞ V∞ To obtain superposition of solution of potential equation for variable sound speed a, the above equation is:- Then linearization is possible and equation (10) becomes, 2 2 2 a γ −1 2 Vx+ V y + V z 2 =1 −M − 1 (4) (1−M ∞ ) φxx + φ yy + φ zz = 0 …(12) ∞ 2 a∞ 2 V ∞ Equation (11) is valid for subsonic and supersonic flow. Flow around the aerofoil in terms of velocity For M ≈ 1, transonic flow equation (9) becomes components:- ∞ (r + 1) −φφxxx ++= φ yy φ zz 0 …(13) V∞ Yadav and Yadav 37 In terms of cylindrical polar coordinates, equation (9) u can be written as C p = − 2 v∞ 112γ − 1 (1−M2 )φφ +++= φ φ M2 φφ 1 + M 2 This fundamental equation is applicable for subsonic ∞ xx rrr rr2 θθ V ∞ x xx 2 ∞ ∞ …(14) and supersonic. Using condition of equation (11), equation (14) becomes IV. RELATION BETWEEN COMPRESSIBLE AND INCOMPRESSIBLE FLOW USING PERTURBATION 2 1 1 THEORY (1−M∞ )φφxx +++ rr φ r 2 φ θθ = 0 r r …(15) Laplace equation for compressible flow is,
Equation (15) is required equation for subsonic and 2 (1−M )φ + φ = 0 …(1) supersonic flow in cylindrical coordinates. ∞ xx zz For transonic flow, equation (14) becomes Laplace equation for incompressible flow is ()φ+ () φ = 0 (2) γ +1 1 1 xx inc. zz inc . … φφ++ φ φ + φ = 0 Vxxx rr r r r 2 qq ∞ …(16) x and z coordinates are along and normal to the flow direction respectively. Equation (16) represents axially symmetric transonic flow. Using boundary conditions, the above equations can be written as, III. PRESSURE COEFFICIENT OF 3D COMPRESSIBLE x= x …(3a) FLOW FOR SUBSONIC &S UPERSONIC FLOW inc
Friction causes flow separation and separation of body zinc= k 1 z …3(b) layer causes adverse pressure gradient which in turn arise pressure drag and overall drag is determined by φ(,)xz= kz φ inc (, x inc. z inc . ) body shape. The difference between local and freestream pressure is known as pressure coefficient. From (1) & (3), 2 2 Pressure distribution can be calculated by velocity 2∂φ 2 ∂ φ . K(1− M )inc. + K inc = 0 …(4) distribution. Flow pattern, inside the circle is not 2∞ 2 1 2 ∂xinc. ∂ z inc . influenced by the flow outside the circle. In irrotational flow, symmetry of pressure distribution shows that no This is similar to incompressible potential equation: drag is experienced by a steadily moving body. Pressure 2 drag can be reduced for any shape by taking separation If K1 =1 − M ∞ …(5) point as far as possible from the leading edge or forward stagnation point. Total pressure at the face of object is Value of K 2 can be find out using boundary conditions. higher than total pressure behind the object. In case of slender bodies, using perturbation theory, we have Pressure coefficient C p, for 3D flow is γ w w dz 2 2 2 γ −1 ≈ = …(6) 2γ − 12 2 u u + v + w C=−1 M + − 1 V∞+ u V ∞ dx p 2 ∞ 2 γ M∞ 2 V∞ V ∞ u But << 1 , the above equation can be written as Where M∞ = free stream Mach number V∞ V = Free stream velocity ∞ ∂z w=∂(φ / ∂ z ) = V …7(a) γ = Ratio of specific heat ∞ ∂x u, v, w = x, y &z components of perturbation velocity dz Expanding binomially and neglecting higher order w=∂(φ / ∂ z ) = V inc . …7(b) inc. inc . inc .0 z inc . = ∞ terms, we get, dx inc .
2 2 2 From equation (3), u2 uvw+ Cp =−2 +− (1 M ∞ ) 2 + 2 Vi V∞ V ∞ (/)∂∂=φz KK (/)0 ∂∂ φ z = …(8) z=0 12 inc . inc . z inc .
For planar bodies, C P is Yadav and Yadav 38
dz dz 1 = K K inc . (9) sin µ = dx1 2 dx M inc . where µ= Mach angle dz 2 dz inc . =K2 1 − M ∞ It is Mach angle – Mach number relation dx dx inc . …(10) From fig. 1 we can conclude that From equation (10), it is clear that profile slope in I. For incompressible medium, M=0 and pressure 2 compressible flow is K2 1− M ∞ times the slope in disturbance moves in all directions. incompressible flow. II. For subsonic speed, M<1, pressure disturbance is felt at every point of space with asymmetrical V. STUDY OF MOVING DISTURBANCE pressure pattern. Disturbance waves travels with local velocity of sound III. In case of supersonic speed M>1, pressure with respect to medium. Moving objects create pressure disturbance apex lies at its apex, included in a cone. disturbance which propagates as shown in fig. below:- Disturbance waves can be considered identical to sound pulses when apex angle is very small. Small deviation in streamlines causes small increase in pressure in Mach cone. Finite deviation causes finite pressure across shock wave. Mach waves and Mach lines are linear and inclined at an angle µ in uniform supersonic flow and varies from point to point in non –uniform supersonic flow resulting Mach lines curved.
Normal, Shock Waves Compression front supersonic flow across which the flow properties jump, and large gradients in
temperature, pressure and velocity occur in thin region, Fig. 2 where transport phenomena of momentum and energy The major difference between subsonic and supersonic are important, is known as Shock. In flow field, flow is, different time taken by disturbance waves. thickness of the shock is comparable to the mean free In subsonic flow, streamline detects the presence of path of the gas molecules. obstacle earlier and flow smoothly around the obstacle. Quantitative Analysis of Normal Shock Wave But in case of supersonic flow, presence of obstacle is For quantitative analysis, let us consider an adiabatic, detected only when hit by it. Moreover obstacle constant area flow through a non-equilibrium region, as becomes source and streamlines are deviated at Mach shown in fig. given below. To define the flowproperties, cone. Surprisingly flow disturbance is sudden and flow let two sections 1 and 2, away from the non-equilibrium behind the object changes quickly. region. Flow for wedge is shown in fig. given below.
Fig. 4. Flow through a normal shock. Fig. 3. Equation of motion for the flow can be written as: From continuity equation, In fig.2(d), it is clear that conical zone is formed by ρV= ρ V …(1) object in supersonic flow, in which object is located at 11 22 the nose and moving object is confined to the cone. No Momentum equation is: 2 2 disturbance is felt by flow field, outside the zone. That is P1+ρ 11 V = P 2 + ρ 22 V …(2) why it is termed as zone of silence. From fig. 2(d), 1 1 Energy equation is: h+ V2 = h + V 2 …(3) 12 1 2 2 2 Yadav and Yadav 39 The above mentioned equations are applicable to all 1 M * = …(9) gases and no restriction on the size, if the reference 2 M * sections 1 and 2 are outside the non-equilibrium region. 1 Relation between the flow parameters at these two According to equation (9), velocity changes across a sections can be obtained by solving these equations. normal shock must be from supersonic to subsonic. Because there is a no restriction on non-equilibrium Hence the Mach number behind a normal shock is region, so it can be idealized as a thin region, as shown always subsonic. Relation between the characteristic Mach number M * and actual Mach number M is given in fig.4(b). Heat traverses the shock wave, so it can by, neither added to nor taken away from the flow; hence 2 2 (γ + 1) M flow process is adiabatic across the shock wave. Heat . M * = …(10) conduction in shock is provided by temperature and (γ − 1)M 2 + 2 velocity gradients. This equation shows that the Mach number behind the For a perfect gas, equation of state is: shock is a function of only the Mach number M 1 ahead P= ρ ST …(4) of the shock, for a perfect gas when M 1 = 1, M 2 =1 ,
Enthalpy is h= Cp T …(5) then it becomes infinitely weak normal shock, identical Now, it is possible to obtain explicit solution in terms of to a Mach wave. Mach wave at µ = sin−1 (1/M ) is
Mach number as – always less than π / 2 in supersonic field. As M 1 is Dividing equation (2) and (1), we have greater than 1, then the normal shock becomes stronger
P1 P 2 and M 2 becomes less than 1, and in this limit, as M 1→∞, − =V2 − V 1 …(6) ρ11V ρ 22 V M →(γ − 1) / 2 γ with γ = 1.4 to 0.378. 2 2 As speed of sound a= γ P / ρ equation (6) becomes V V V 2 1= 1 = 1 = M * …(11) 2 2 *2 1 a a V2 VV 1 2 a 1− 2 =V − V …(7) 2 1 γV1 γ V 2 Using energy equations, we get The ratio of velocities is, 2 2 2 2 2 2 ρ2V 1( γ + 1) M 1 V1 a 1 V 2 a 2 1γ + 1 * + =+ = a and = = 2 …(12) 2γ− 12 γ − 121 γ − ρ1VM 2( γ − 1) 1 + 2 2 2 Pressure relation can be obtained as a1 and a2 can be expressed as: PP− =ρ V2 − ρ V 2 γ+1 γ − 1 2 1 11 22 a2= a *2 − V 2 From equation (1) 12 2 1 P− P =ρ VV( − V ) γ+1 γ − 1 2 1 111 2 a2= a *2 − V 2 2 2 2 2 2 V =ρ V 1 − 2 As the flow process across the shock wave is adiabatic, 1 1 V1 * 2 2 in a in the above relations for a1 and a2 has same Dividing above equation by P1, we get constant value. PP− ρ V2 V Substituting these relations in equation (7), we get 21= 111 − 2 P P V γ+1a*2 γ − 1 γ + 1 a *2 γ − 1 1 1 1 2 −V1 − + VVV 221 =− as a1= (γ P 1 ) / ρ 1 , we get 2γγV1 2 2 γγ V 2 2 P− P V γ+1*2 γ − 1 2 1=γ M 2 1 − 2 …(13) ()VVa21− + ( VVVV 2121 −=− ) 1 2γV V 2 γ P1 V 1 1 2 Dividing above equation by V− V , we get V 2 1 Putting value of 2 in above equation, we obtain γ+1 γ − 1 V1 a*2 + = 1 2γV V 2 γ P− P2 + (γ − 1) M 2 1 2 . 2 1 2 1 …(14) =γ M1 1 − 2 *2 P1 (γ + 1) M1 or a= V1 V 2 …(8) which is called the Prandtl relation. The above equation can be written as, P 2γ 2 * V 2 =1 + (M 2 − 1) …(15) In terms of the speed ratio M = * , equation (8) can a P1 γ +1 be written as:
Yadav and Yadav 40 P− P ∆P oblique shocks are generated in the engine intakes of ratio of 2 1 = is called the shock strength. P P supersonic flight vehicles, where the engine has 1 1 provision to control its backpressure. When the The entropy change in terms of pressure and backpressure increases upto a definite value, the oblique temperature ratios across the shock can be written as shock at the engine inlet would becomes a strong shock T2 P 2 and decelerate the supersonic flow passing through it to SSC2− 1 =p ln − R ln T1 P 1 subsonic level. 2 2 2 Dependence of deflection angle θ on Mach number M 1 Tha2222(γ+ 1)( γ MM 11 + 1)( − 1) as = =2 =+1 2 2 and shock angle β is, Tha111(γ + 1) M 1 M 2sin 2 β − 1 2 tanθ= 2cot β 1 2(γ− 1) γ (M1 + 1) 2 2 SSC−=+ln 1 ( M −− 1) M1 (γ+ cos 2 β ) + 2 2 1 p (γ + 1) 2M 2 1 1 This θ− β − M relation plays important role ion 2γ 2 analysis of oblique shocks. Rln 1+ ( M 1 + 1) …(16) γ +1 The expression on R.H.S. of above equation becomes The above equation shows that for a perfect gas with a 1 zero at β= π / 2 and β = sin −1 , which are the given value of V, all variables are functions of M 1 only. M This explains the importance of Mach number in the 1 quantitative governance of compressible flows flow limiting value of β. In this range, deflection angle θ is properties changes within a very short distance across positive and have a maximum value. For γ = 1.4, results the shock. This distance may be order of 10 −5 cm. Hence obtained from above equation are plotted as: the value of velocity and temperature gradients are very large within the shock structure. Hence entropy across the shock changes due to large gradients. If the shock wave is not stationary, than neither the total enthalpy nor the total temperature are constant across the shock wave. Entropy varies with loss in pressure. In this case it becomes independent of velocity and there is nothing like stagnation entropy, so the entropy difference without any reference to the velocity level is,
P01 S2− S 1 = R ln …(17) P02 and −1/(γ − 1) γ/( γ − 1) PM2γ (γ + 1) 2 01 =1 + (M 2 − 1) 1 …(18) 1 2 P02 γ +1 (γ − 1)M1 + 1 It connects the stagnation pressure to the normal shock to flow Mach number ahead of the shock. In supersonic flow, when a pilot probe is placed, then a detached shock standing ahead of probe nose. Total pressure Fig. 5. Oblique Shock Solution. behind the detached shock is measured by the probe. We can find the flow Mach number ahead of the shock from equation (18), by knowing the stagnation pressure ahead of the shock, which is the pressure in the reservoir, for isentropic flow up to the shock. Oblique Shock Oblique shocks are essentially compression fronts across which the flow decelerates and the static pressure, Fig. 6. Detached shocks. temperature and density varies upto higher values. In deceleration, Mach number behind the shock is greater than unity, then the shock is termed weak oblique shock. 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