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Potential Equations and Pressure Coefficient For 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 xinc = x …(3a) FLOW FOR SUBSONIC &S UPERSONIC FLOW 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 ∞ (/)∂∂=φzz KK (/)0 ∂∂ φ inc z inc z = …(8) =0 12 .
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