Axial Compressor Design Onur Tuncer Istanbul Technical University Faculty of Aeronautics and Astronautics Department of Aeronautical Engineering Maslak, Istanbul 34469 [email protected] November 23, 2011 Outline Introduction Thermodynamics Fluid Mechanics Axial-Flow Compressor Blade Profiles 2-D Blade-to-Blade Flow O. Tuncer (ITU) Axial Compressor November 2011 2 / 148 Introduction Axial Flow Compressor Basics Axial vs. Centrifugal Compressors O. Tuncer (ITU) Axial Compressor November 2011 3 / 148 Introduction Axial Flow Compressor Basics Construction of an Axial Compressor A stationary row of blades (stator) is followed by a rotating row of blades (rotor). A compressor stage is made up of a rotor and a stator. O. Tuncer (ITU) Axial Compressor November 2011 4 / 148 Introduction Basic Velocity Diagrams for a Stage Polar Surface View of a Stage The rotor row is rotating with a velocity U = ωr Viewed in a reference frame rotating with the rotor, the upstream velocity W is called the relative velovity. The rotor deflects the flow such that the velocity in the stationary frame of reference of the stator (the absolute velocity), C is properly aligned to enter the stator row. O. Tuncer (ITU) Axial Compressor November 2011 5 / 148 Introduction Basic Velocity Diagrams for a Stage Shrouded Stator Blades O. Tuncer (ITU) Axial Compressor November 2011 6 / 148 Introduction Basic Velocity Diagrams for a Stage Guide Vane Velocity Triangles O. Tuncer (ITU) Axial Compressor November 2011 7 / 148 Introduction Basic Velocity Diagrams for a Stage Velocity Triangle for a Rotor O. Tuncer (ITU) Axial Compressor November 2011 8 / 148 Introduction Basic Velocity Diagrams for a Stage Velocity Triangle Calculations Wθ = Cθ − ωr The axial components of velocity are identical in both reference frames. W z = Cz The absolute and relative velocities are, q 2 C = Cz + Cθ2 q W = Cz2 + Wθ2 The absolute β and relative β 0 flow angles are, tan β = Cθ /Cz tan β 0 = Wθ /Cz O. Tuncer (ITU) Axial Compressor November 2011 9 / 148 Introduction Similitude and Performance Characteristics Similarity Two turbomachines are completely similiar if the ratios of all corresponding length dimensions, velocity components and forces are equal. Equivalent Flow Rate Parameter Q = ṁ/ρt Local Axial Flow Velocity Q = ṁ/ρt Volume Flow Machines ρ0 /ρt0 is a function of Cz 0. Therefore, unique velocity diagrams are associated with a unique Q0 /A0 , yet can correspond to many values of ṁ/A0 . O. Tuncer (ITU) Axial Compressor November 2011 10 / 148 Introduction Similitude and Performance Characteristics An Equivalent Performance Map O. Tuncer (ITU) Axial Compressor November 2011 11 / 148 Introduction Similitude and Performance Characteristics More on Similarity I True equivalent performance is obtained if working fluids obey the perfect gas equation. I Similarity is also compromised if the compressors operate at substantially different Reynolds numbers. O. Tuncer (ITU) Axial Compressor November 2011 12 / 148 Introduction Similitude and Performance Characteristics More on Similarity Alternative Equivalent Flow Parameters √ √ Q/at0 ∝ ṁ Tt0 /Pt0 ∝ ṁ θ/δ Sound √ of speed is calculated by, a = kRT θ and δ relate inlet total conditions to some reference condition (most often to standart atmospheric conditions). θ = Tt0 /Tref δ = Pt0 /Pref The equivalent speed can be replaced by, N/at0 ∝ N/ O. Tuncer (ITU) p √ Tt0 ∝ N/ θ Axial Compressor November 2011 13 / 148 Introduction Similitude and Performance Characteristics Efficiency Previous figure only shows part of the information. However, it is important to know how much work is necessary to drive the compressor. η = ∆Hrev /∆H Z ∆Hrev = rev O. Tuncer (ITU) Axial Compressor dP ρ November 2011 14 / 148 Introduction Similitude and Performance Characteristics An Equivalent Efficiency Map O. Tuncer (ITU) Axial Compressor November 2011 15 / 148 Introduction Stage Matching and Stability Stage Matching and Stability I Each blade row achieves best performance for a specific inlet flow angle, where losses are minimum. I The designer seeks to ”match” succeeding blade rows such that all operate close to their optimum inlet flow angles, at a specific operating condition (i.e. design point or match point) I At lower flow rates the characteristic has a positive slope which is theoretically unstable. This severe unstable operation is commonly called as surge. I In other cases abrupt stall might occur. I For higher flow rates no rise in pressure might occur. This is called choke. O. Tuncer (ITU) Axial Compressor November 2011 16 / 148 Introduction Dimensionless Parameters Dimensionless Parameters Euler Turbine Equation ∆H = U(Cθ2 − Ctheta1 ) Total Enthalpy 1 H = h + C2 2 O. Tuncer (ITU) Axial Compressor November 2011 17 / 148 Introduction Dimensionless Parameters Dimensionless Parameters Stage Work Coefficient Ψ = ∆H/U 2 = (Cθ2 − Ctheta1 )/U Stage Flow Coefficient φ = Cz 1/U Stage Reaction R = (h2 − h1 )/(h3 − h1 ) O. Tuncer (ITU) Axial Compressor November 2011 18 / 148 Introduction Dimensionless Parameters Relationship Between φ, Ψ, R and Velocity Diagrams 50% Reaction Stages tan β10 = −(Ψ/2 + R)/φ tan β20 = (Ψ/2 − R)/φ tan β1 = (1 − R − Ψ/2)/φ tan β2 = (1 − R + Ψ/2)φ O. Tuncer (ITU) Axial Compressor November 2011 19 / 148 Thermodynamics First and Second Laws of Thermodynamics First Law of Thermodynamics I Steady-state I Steady-flow I Open system 1 2 q̇ + ẇ = ṁ∆ u + C + P/ρ 2 Note that, h ≡ u + P/ρ O. Tuncer (ITU) Axial Compressor November 2011 20 / 148 Thermodynamics First and Second Laws of Thermodynamics Second Law of Thermodynamcis Specific Entropy ds = dqrev T Second Law ∆s ≥ 0 Fundamental Thermodynamic Equation for Entropy Tds = dh − VdP O. Tuncer (ITU) Axial Compressor November 2011 21 / 148 Thermodynamics Efficiency An Enthalpy Entropy Diagram O. Tuncer (ITU) Axial Compressor November 2011 22 / 148 Thermodynamics Efficiency Polytropic Efficiency I Polytropic efficiency is also known as small stage efficiency or true aerodynamic efficiency. I Instead of using a path of constant entropy as the reversible path, polytropic efficiency uses as path of constant efficiency defined by ηP = ρ1 dP dh . Polytropic Head ∆Hp = ∆H − (sd − si )(Ttd − Tti )/ ln(Ttd /Tti ) Total to Polytropic Efficiency ηP = O. Tuncer (ITU) ∆Hp ∆H Axial Compressor November 2011 23 / 148 Thermodynamics Fluid Equation of State Fundamentals Fundamental Relations Thermal Equation of State P = P(ρ, T ) Calorific Equation of State h = h(T , P) u = u(T , P) O. Tuncer (ITU) Axial Compressor November 2011 24 / 148 Thermodynamics Fluid Equation of State Fundamentals Ideal Gas Law P = ρRT R = Ru /M Ru = 8314 Pa.m3 /(kmol.K ) O. Tuncer (ITU) Axial Compressor November 2011 25 / 148 Thermodynamics Fluid Equation of State Fundamentals A Pressure Enthalpy Diagram Schematic O. Tuncer (ITU) Axial Compressor November 2011 26 / 148 Thermodynamics The Calorific Equation of State The Calorific Equation of State ∂h◦ = ∂T ◦ P ∂h cv◦ (T ) = ∂T V cp◦ (T ) For a thermally perfect gas, cp◦ (T ) − cv◦ (T ) = R ◦ ◦ T Z h (T ) = h (Tref ) + u ◦ (T ) = u ◦ (Tref ) + Tref T Z cp◦ (T )dT cv◦ (T )dT Tref O. Tuncer (ITU) Axial Compressor November 2011 27 / 148 Thermodynamics Entropy and the Speed of Sound Entropy and the Speed of Sound Specific Entropy ◦ ◦ Z T s (T , P) = s (Tref , Pref ) + Tr ef cp◦ (T ) − R ln(P/Pref ) T Specific Entropy for a Calorifically Perfect Gas s ◦ (T , P) = s ◦ (Tref , Pref ) + cp◦ ln(T /Tref ) − R ln(P/Pref ) O. Tuncer (ITU) Axial Compressor November 2011 28 / 148 Thermodynamics Entropy and the Speed of Sound Speed of Sound Thermodynamic Relation for the Speed of Sound ◦ a = ∂P ∂ρ =k s ∂P ∂ρ T Ratio of Specific Heats k = cp /cv Speed of Sound for a Thermally Perfect Gas a◦ = O. Tuncer (ITU) √ kRT Axial Compressor November 2011 29 / 148 Thermodynamics The Thermal Equation of State for Real Gases The Thermal Equation of State for Real Gases General Thermal Equation of State for a Real Gas P/(ρRT ) = z(T , P) where z is the compressibility factor. For ideal gases z = 1. Parametric Equations Simple two-parameter equations of state are a good choice for general aerothermodynamic design and analysis. O. Tuncer (ITU) Axial Compressor November 2011 30 / 148 Thermodynamics The Thermal Equation of State for Real Gases Redlich-Kwong Equation of State Redlich-Kwong Equation P= a RT √ − V − b V (V + b) Tr where Tr = T /Tc is the reduced temperature and, a = 0.42747R 2 Tc /Pc b = 0.08664RTc /Pc O. Tuncer (ITU) Axial Compressor November 2011 31 / 148 Thermodynamics Thermodynamic Properties of Real Gases Thermodynamic Properties of Real Gases Departure Functions I Specification of the calorific equation of state, h◦ , u ◦ are limited to state points where the fluid is thermally perfect. I For non-ideal fluids h and u are functions of P and T . I Thermodynamics properties of a non-ideal fluid are best accomplished utilizing departure functions. I Departure functions are defined as the difference between the actual value of a parameter and its value under conditions where the fluid is thermally perfect. O. Tuncer (ITU) Axial Compressor November 2011 32 / 148 Thermodynamics Thermodynamic Properties of Real Gases Departure Functions Corresponding Specific Volume V ◦ = RT /P ◦ If A=Helmholtz Energy A − A◦ = − Z V (P − RT /V )dV − RT ln(V /V ◦ ) ∞ ◦ s −s =− ∂(A − A◦ ) ∂T V h − h◦ = (A − A◦ ) + T (s − s ◦ ) + RT (z − 1) u − u ◦ = (A − A◦ ) + T (s − s ◦ ) O. Tuncer (ITU) Axial Compressor November 2011 33 / 148 Thermodynamics Thermodynamic Properties of Real Gases Redlich-Kwong Departure Functions h−h ◦ s − s◦ a V +b −n = PV − RT − (n + 1)Tr ln b b V V −b+c na −n V +b = −R ln − T ln V◦ V bT r b where, c = 0 and n = 0.5 for the original Redlich-Kwong equation of state. O. Tuncer (ITU) Axial Compressor November 2011 34 / 148 Thermodynamics Thermally and Calorifically Perfect Gases Thermally and Calorifically Perfect Gases When the fluid can be considered thermallyperfect (z = 1) and calorifically perfect (cp , cv and k are constants), equation of state calculations are greatly simplified. Calculations for Enthalpy and Entropy h = href + cp (T − Tref ) s = sref + cp ln(T /Tref ) − R ln(P/Pref ) O. Tuncer (ITU) Axial Compressor November 2011 35 / 148 Thermodynamics Thermally and Calorifically Perfect Gases Thermally and Calorifically Perfect Gases Relation Between Total and Static Conditions 1 cp (Tt − T ) = C 2 2 T /Tref = (P/Pref ) k−1 k = (ρ/ρref )k−1 Efficiency Calculations k−1 ηad = ηp = O. Tuncer (ITU) (Ptd /Pti ) k − 1 Ttd /Tti − 1 k − 1 ln(Ptd /Pti ) k ln(Ttd /Tti ) Axial Compressor November 2011 36 / 148 Thermodynamics The Pseudo-Perfect Gas Model The Pseudo-Perfect Gas Model The concept is to use fictitious values of cp , cv and k in an otherwise standart calorifically and thermally perfect gas model. √ R̄ = R z1 z2 c¯p = (h2 − h1 )/(T2 − T1 ) c¯v = (u2 − u1 )/(T2 − T1 ) k̄ = ln(P2 /P1 )/ ln(ρ2 /ρ1 ) O. Tuncer (ITU) Axial Compressor November 2011 37 / 148 Thermodynamics Component Performance Parameters Diffuser Diffuser Efficiency ηdiff = ∆had ∆h Pressure Recovery Coefficient cp = O. Tuncer (ITU) Pd − Pi Pti − Pi Axial Compressor November 2011 38 / 148 Thermodynamics Component Performance Parameters Nozzle Pressure Loss Coefficient Nozzle Efficiency ηnoz = ω̄ = Cd2 − Ci2 2 − C2 Cad i ηnoz = 1 − ∆Pt 1 2 2 ρC Pti − Ptd Pi − Pd Pressure Loss ∆Pt Pti − Ptd = = T ∆s ρ ρ O. Tuncer (ITU) Axial Compressor November 2011 39 / 148 Thermodynamics Gas Viscosity Gas Viscosity Dean and Stiel Model 1/6 ξ = Tc / √ 2/3 MPc Low pressure fluid viscosity, 8/9 µ0 ξ = (3.4.10−4 Tr , Tr ≤ 1.5 µ0 ξ = 0.001668(0.1338Tr − 0.0932)5/9 , Tr > 1.5 Then, the viscosity at any pressure can be defined by the following departure function. (µ − µ◦ )ξ = (1.08.10−4 )[exp(1.439ρr ) − exp(−1.111ρr1.858 )] O. Tuncer (ITU) Axial Compressor November 2011 40 / 148 Fluid Mechanics Flow in a Rotating Coordinate System Flow in a Rotating Coordinate System The analysis of the flow in the rotor blade rows is accomplished in a coordinate system that rotates with the blade. Wθ = Cθ − ωr The axial and radial velocity components are independent of rotation. Wz = Cz Wr = Cr Meridional Velocity Component Wm = O. Tuncer (ITU) q Wz2 + Wr2 = Cm Axial Compressor November 2011 41 / 148 Fluid Mechanics Flow in a Rotating Coordinate System Stream Surface and Natural Coordinate System A stream surface is defined as a surface having no fluid velocity component normal to it. Schematic of a Stream Surface Natural Coordinate System (dm)2 = (dr )2 + (dz 2 ) O. Tuncer (ITU) Axial Compressor November 2011 42 / 148 Fluid Mechanics Flow in a Rotating Coordinate System Euler Turbine Equation Consider the flow through a thin stream sheet (i.e. a thin annular passage bounded by two stream surfaces). The torque τ acting on the fluid between merional stations 1 and 2 is provided by the conservation of angular momentum. τ = ṁ(r2 Cθ2 − r1 Cθ1 ) The torque must balance the power input. ẇ = ωτ = ω ṁ(r2 Cθ2 − r1 Cθ1 ) Euler Turbine Equation H2 − H1 = ω(r2 Cθ2 − r1 Cθ1 ) O. Tuncer (ITU) Axial Compressor November 2011 43 / 148 Fluid Mechanics Flow in a Rotating Coordinate System Rothalpy Total enthalpy change is produced by a transfer of mechanical energy between the fluid and the rotating blade row. I = H − ωrCθ For a stationary blade row, I = H. Aerodynamic analysis of axial compressors involve the solution of conservation equations in both rotating (rotors) and stationary (stators) coordinates. The relationship between relative total enthalpy H 0 in a rotating, and absolute total entalhalpy in a stationary coordinate system. 1 1 h = H0 − W 2 = H − C 2 2 2 O. Tuncer (ITU) Axial Compressor November 2011 44 / 148 Fluid Mechanics Flow in a Rotating Coordinate System Rothalpy (Continued) The relative velocity W , W = q Wm2 + Wθ2 Total enthalpies, 1 1 H 0 = H − ωrCθ + (ωr )2 = I + (ωr )2 2 2 Since I is constant on the stream surface, above equation allows the calculation of H 0 at all points on a stream surface when one value is known, e.g. at the inlet. O. Tuncer (ITU) Axial Compressor November 2011 45 / 148 Fluid Mechanics Adiabatic Inviscid Compressible Flow Adiabatic Inviscid Compressible Flow Vector Form of Momentum Equation in a Rotating Coordinate Frame ~ ~ 1~ DC DW ~ )+ω = − ∇P + 2(~ ωx W ~ x(~ ω x~r ) = Dt ρ Dt where, ~ ~ DW ∂W ~ .∇) ~ ~ W = + (W Dt ∂t Hence the momentum equation in rotating coordinates is, ~ ~ ∂W ∇P ~ .∇) ~ + 2(~ ~ )+ω ~ W + (W ωx W ~ x(~ ω x~r ) = ∂t ρ O. Tuncer (ITU) Axial Compressor November 2011 46 / 148 Fluid Mechanics Adiabatic Inviscid Compressible Flow Vector Form of Continuity and Energy Equations ∂ρ ~ ~ )=0 + ∇.(ρW ∂t 1 ∂P ∂I ~ .∇)I ~ =0 − + (W ∂t ρ ∂t O. Tuncer (ITU) Axial Compressor November 2011 47 / 148 Fluid Mechanics Adiabatic Inviscid Compressible Flow Governing Equations in Natural Coordinates ∂ρ 1 ∂r ρWm ∂ρWθ + + + κn ρWm = 0 ∂t r ∂m ∂θ Wm Wθ ∂Wm sin φ 1 ∂P ∂Wm + Wm + − [Wθ + ωr ]2 = − ∂t ∂m r ∂θ r ρ ∂m ∂Wθ ∂Wθ Wθ ∂Wθ Wm sin φ 1 ∂P + Wm + + [Wθ + 2ωr ] = − ∂t ∂m r ∂θ r r ρ ∂θ κm Wm2 + cos φ 1 ∂P [Wθ + ωr ]2 = r ρ ∂n ∂I 1 ∂P ∂I Wθ ∂I − + Wm + =0 ∂t ρ ∂t ∂m r ∂θ O. Tuncer (ITU) Axial Compressor November 2011 48 / 148 Fluid Mechanics Adiabatic Inviscid Compressible Flow Governing Equations in Natural Coordinates The curvature of the stream sheet κm and the normal of the surface κn are related to the angle φ as follows. ∂φ ∂m ∂φ 1 ∂b = ∂n b ∂m κm = − κn = Parameter b is the thickness of the stream sheet bounded by two stream surfaces. O. Tuncer (ITU) Axial Compressor November 2011 49 / 148 Fluid Mechanics Adiabatic Inviscid Compressible Flow Applications Adiabatic Inviscid Compressible Flow Applications I To determine the flow in the meridional plane I Blade-to-blade flow solutions I Hub to shroud flow solutions I Quasi three dimensional analysis O. Tuncer (ITU) Axial Compressor November 2011 50 / 148 Fluid Mechanics Boundary Layer Analysis Boundary Layer Analysis The basic premise of boundary layer theory is that the viscous effects are confined to a thin layer close to the physical surfaces bounding the flow passages. I I Blade surface boundary layers play an important role in viscous losses and stall or boundary layer seperation. Endwall boundary layers can produce substantial viscous blockage effects that significantly affect compressor performance. O. Tuncer (ITU) Axial Compressor November 2011 51 / 148 Fluid Mechanics Two-Dimensional Boundary Layer Analysis Two-Dimensional Boundary Layer Analysis I Two-dimensional boundary layer analysis is a useful approximation in blade design. I Two dimensional blade sections designed between the hub and tip are stacked together to create the actual three dimensional compressor blade. Boundary Layer Equations Basic conservation of mass and momentum provide the governin equations for two dimensional boundary layer flow over an adiabatic wall. ∂ρbu ∂ρbv + =0 ∂x ∂y u O. Tuncer (ITU) ∂u ∂u 1 ∂P 1 ∂τ +v + = ∂x ∂y ρ ∂x ρ ∂y Axial Compressor November 2011 52 / 148 Fluid Mechanics Two-Dimensional Boundary Layer Analysis Integral Form of Boundary Layer Analysis Integrating the conservation of mass across the boundary layer and applying the Liebnitz rule to interchange the order of integration and differentiation yields. ∂ ∂x Z δ bρudy = bρe ue 0 ∂δ ∂ − bρe ve = [bρe ue (δ − δ ∗ )] ∂x ∂x Displacement Thickness ∗ Z δ [ρe ue − ρu]dy ρe ue δ = 0 It is a fictitious thickness used to correct the mass balance relative to the inviscid flow solution. O. Tuncer (ITU) Axial Compressor November 2011 53 / 148 Fluid Mechanics Two-Dimensional Boundary Layer Analysis Integral Form of Boundary Layer Analysis (Continued) Momentum Thickness ρe ue2 θ = Z ρu[ue − u]dy Combining displacement and momentum thicknesses yields, Z δ ρu 2 dy = ρe ue2 [δ − δ ∗ − θ] 0 If the free stream conditions are applied within the boundary layer with no flow in the thickness δ ∗ , and no momentum in the thickness θ, momentum conservation will be corrected for viscous effects. This is the basis of integral boundary layer analysis method. O. Tuncer (ITU) Axial Compressor November 2011 54 / 148 Fluid Mechanics Two-Dimensional Boundary Layer Analysis Integral Form of Boundary Layer Analysis (Continued) Integrating the momentum equation across the boundary layer. Further noting that P = Pe across the boundary layer, ∂ ∂x Z 0 δ bρu 2 dy − ρe ue2 ∂δ ∂Pe + ρe ue ve + δ = −τw ∂x ∂x Arranging and manipulating this statement one arrives at the well-known momentum integral equation. 1 ∂bρe ue2 θ ∂ue + δ ∗ ρe ue = τw b ∂x ∂x O. Tuncer (ITU) Axial Compressor November 2011 55 / 148 Fluid Mechanics Two-Dimensional Boundary Layer Analysis Flow Entrainment into the Boundary Layer The momentum integral equation is valid for both laminar and turbulent boundary layers. Laminar boundary layer analysis usually employs specific boundary layer flow profile assumptions to permit direct integration of the momentum integral equation. Turbulent boundary layer analysis usually employs several empirical models for solution, which may include specific boundary layer flow profile assumptions. Usually turbulent boundary layer analysis employs a second conservation equation (i.e mass, energy, moment of momentum). ∂ [bρe ue (δ − δ ∗ )] = bρe ue E ∂x Entrainment Function E= O. Tuncer (ITU) ∂δ ve − ∂x ue Axial Compressor November 2011 56 / 148 Fluid Mechanics Two-Dimensional Boundary Layer Analysis Axisymmetric Three-Dimensional Boundary Layer Analysis The governing equations for axisymmetric three-dimensional boundary layer flow in a rotating coordinate system in natural coordinates are, 1 ∂ρWm ∂ρWy + =0 r ∂m ∂y ∂Wm ∂Wm sin φ 1 ∂Pe ∂τm 2 Wm + Wy − (Wθ + ωr ) = fm − − ∂m ∂y r ρ ∂m ∂y ∂Wθ ∂Wθ sin φ 1 ∂τθ Wm + Wy + Wm (Wθ + 2ωr ) = fθ − ∂m ∂y r ρ ∂y Body force terms, fme = ρe Wme fθe = ρe Wme O. Tuncer (ITU) ∂Wme ∂Pe sin φ + − ρe (Wθe + ωr )2 ∂m ∂m r ∂Wθe sin φ ρe Wme ∂rCthetae + ρe Wme (Wθe + 2ωr ) = ∂m r r ∂m Axial Compressor November 2011 57 / 148 Fluid Mechanics Two-Dimensional Boundary Layer Analysis Integral Form of Axisymmetric Three Dimensional Boundary Layer Equations Boundary layer equations are converted into integral form in the same manner as described earlier. The resulting integral equations are, ∂ [r ρe Wme (δ − δ1∗ )] = r ρe We E ∂m ∂ ∂Wme 2 [r ρe Wme θ11 ] + δ1∗ r ρe Wme ∂m ∂m − ρe Wθe sin φ[Wθe (δ2∗ + θ22 ) + 2ωr δ2∗ = r [τmw + fme vm ] ∂ 2 ∂Wθe ∗ [r ρe Wme Wθe θ12 ] + r δ1 ρe Wme r + sin φ(Wθe + 2r ω) ∂m ∂m = r 2 [τθw + fθe vθ ] O. Tuncer (ITU) Axial Compressor November 2011 58 / 148 Fluid Mechanics Two-Dimensional Boundary Layer Analysis Mass, Momentum and Force Defects ρe Wme δ1∗ Z δ (ρe Wme − ρWm )dy = 0 2 ρe Wme θ11 = Z δ ρWm (Wme − Wm )dy 0 Z δ ρWm (Wθe − Wθ )dy ρe Wme Wθe θ12 = 0 ρe Wθe δ2∗ Z δ = (ρe Wθe )dy 0 2 ρe Wθe θ22 Z δ ρWθ (Wθe − Wθ )dy = 0 Z vm fme δ (fme − fm )dy = 0 Z vθ fθe O. Tuncer (ITU) δ (fθe − fθ )dy = 0 Axial Compressor November 2011 59 / 148 Axial-Flow Compressor Blade Profiles Basic Airfoil Geometry Blade Camber Angle θ = χ1 + χ2 O. Tuncer (ITU) Axial Compressor November 2011 60 / 148 Axial-Flow Compressor Blade Profiles Cascade Nomenclature Cascade Nomenclature Cascade Solidity σ = c/s Stagger Angle γ O. Tuncer (ITU) Axial Compressor November 2011 61 / 148 Axial-Flow Compressor Blade Profiles Cascade Nomenclature Cascade Nomenclature (Continued) κ1 and kappa2 The angles between slopes to the camberline and the axial direction, at the leading and trailing edges respectively. Incidence Angle i = β 1 − κ1 Deviation Angle δ = β2 − κ2 Angle of Attack α = β1 − γ O. Tuncer (ITU) Axial Compressor November 2011 62 / 148 Axial-Flow Compressor Blade Profiles Cascade Nomenclature Notes on Cascade Nomenclature I This cascade nomenclature is directly applicable to blades based on well-defined camberlines such as the circular arc and parabolic arc camberlines (typical of British practice). I American practice is based on NACA airfoils, which typically have infinite camberline slopes at the leading and trailing edges. A suitable approximate reference is needed to define κ, χ, θ, i and δ. Common practice is to use an equivalent circular arc camberline as a reference. I Construction of blades from the base camberline and profile is occasionally a source of confusion. When imposing a profile on a blade with camber, the thickness distribution data should be interpreted in terms of dimensionless distance along the camberline rather than along the chord line. O. Tuncer (ITU) Axial Compressor November 2011 63 / 148 Axial-Flow Compressor Blade Profiles Cascade Nomenclature Dimensionless Data for Axial Flow Compressor Blades O. Tuncer (ITU) Axial Compressor November 2011 64 / 148 Axial-Flow Compressor Blade Profiles NACA 65-Series Profile NACA 65-Series Profile I The NACA 65-series blades are derived from NACA aircraft wing airfoils for approximately uniform loading. I NACA 65-series airfoils are designated by their lift coefficients and maximum thickness to chord ratio. The lift coefficient in tenths first appear in parentheses followed by the thickness to chord ratio. O. Tuncer (ITU) Axial Compressor November 2011 65 / 148 Axial-Flow Compressor Blade Profiles NACA 65-Series Profile NACA 65-12 and Equivalent Circular Arc Camberline Profiles Relation between the effective camber angle and the lift coefficient. tan(θ/4) = 0.1103Cl0 O. Tuncer (ITU) Axial Compressor November 2011 66 / 148 Axial-Flow Compressor Blade Profiles Circular Arc Camberline Circular Arc Camberline Commonly used in conjunction with the British C.4 profile. Also the camberline used for double circular arc profile. O. Tuncer (ITU) Axial Compressor November 2011 67 / 148 Axial-Flow Compressor Blade Profiles Circular Arc Camberline Circular Arc Camberline (Continued) c/2 = Rc (sin(θ/2) yc = −Rc cos(θ/2) y = yc + p Rc − x 2 2y (0)/c = [1 − cos(θ/2)]/ sin(θ/2) = tan(θ/4) O. Tuncer (ITU) Axial Compressor November 2011 68 / 148 Axial-Flow Compressor Blade Profiles Parabolic Arc Camberline Parabolic Arc Camberline Parabolic arc camberline is also used with British C.4 profile and with others as well. The parabolic arc allows a more general blade loading style than the circular arc. Front, mid and rear loaded blades are all possible. This loading can be changed by changing the location of maximum camber. O. Tuncer (ITU) Axial Compressor November 2011 69 / 148 Axial-Flow Compressor Blade Profiles Parabolic Arc Camberline Parabolic Arc Camberline (Continued) The point of maximum camber is located at x = a, y = b. The basic constraints are, y (0) = 0 y (c) = 0 y (a) = 0 y 0 (a) = 0 The camberline is generated using the general second-order equation. √ Ax 2 + 2 AE xy + By 2 + Cx + Dy + E = 0 Note that one of the coefficients is arbitrary. x2 + O. Tuncer (ITU) c − 2a (c − 2a)2 2 c 2 − 4ac xy + y − cx − y =0 b 4b 2 4b Axial Compressor November 2011 (1) 70 / 148 Axial-Flow Compressor Blade Profiles Parabolic Arc Camberline Parabolic Arc Camberline (Continued) To evaluate the blade angles the derivative of the previous expression at x = 0 and x = c can be used. tan χ1 = 4b/(4a − c) tan χ2 = 4b/(3c − 4a) Parabolic arc camberline in terms of camber and the ratio a/c q b/c = [ 1 + (4 tan θ)2 [a/c − (a/c)2 − 3/16] − 1]/(4 tan θ) Note that, 0.25 < a/c < 0.75s O. Tuncer (ITU) Axial Compressor November 2011 71 / 148 Axial-Flow Compressor Blade Profiles British C.4 Profile British C.4 Profile I One of the several profiles in the British C series. I With respect to NACA-65 series, C.4 is thicker towards the leading edge. I Maximum thickness at 30% chord. I Less effective at higher Mach numbers but higher structural integrity. I C.7 profile has more use in compressors and quite similiar to NACA-65 series. O. Tuncer (ITU) Axial Compressor November 2011 72 / 148 Axial-Flow Compressor Blade Profiles British C.4 Profile Designation of C Series Profiles I C-series profiles are designated by a code giving tb , profile, θ, camberline and a/c. I 10C4/20P40 is a 10% thick C.4 profile with 20◦ camber angle using a parabolic arc camberline with a/c = 0.4. I 10C4/20C50 similiar but with a circular arc camberline. Note: Well established empirical performance prediction models exist for C.4 profiles. O. Tuncer (ITU) Axial Compressor November 2011 73 / 148 Axial-Flow Compressor Blade Profiles Double Circular Arc Profile Double Circular Arc Profile Double circular arc profiles are constructed with both surfaces formed by circular arcs, that blend with a nose radius r0 applied both at the leading and trailing edgrs. ∆xU = (RU − r0 ) sin(θU /2) = c/2 − r0 cos(θ/2) O. Tuncer (ITU) Axial Compressor November 2011 74 / 148 Axial-Flow Compressor Blade Profiles Double Circular Arc Profile Double Circular Arc Profile (Continued) ∆yU = Ru − y (0) − tb /2 + r0 sin(θ/2) = RU − d d = y (0) + tb /2 − r0 sin(θ/2) The Phythagorean theorem applied to the right triangle requires, [RU − r0 ]2 = [RU − d]2 + [c/2 − r0 cos(θ/2)]2 Ru = d 2 − r02 + [c/2 − r0 cos(θ/2)]2 2(d − r0 ) The leading and trailing edge radii are constructed about their centers at y = r0 sin(θ/2) and x = ±[c/2 − r0 cos(θ/2)] to blend with the circular arc. O. Tuncer (ITU) Axial Compressor November 2011 75 / 148 Axial-Flow Compressor Blade Profiles NACA A4 K6 63-Series Guide Vane Profile NACA A4 K6 63-Series Guide Vane Profile This vane has excellent flow guidance and a wide incidence operating range. The camberline is developed by combining a front-loaded A profile with Cl0 = 0.4 and a uniform loaded K profile with Cl0 = 0.6, which is designated as the A4 K6 camberline corresponding to Cl0 = 1. This geometry is combined with the 6% thick NACA-63 series profile as the base guide vane geometry. Similiar to 65 series blades, the camberline coordinates can be scaled directly by lift cooefficient to alternate camberlines. The thickness distribution can be scaled as well. O. Tuncer (ITU) Axial Compressor November 2011 76 / 148 Axial-Flow Compressor Blade Profiles NACA A4 K6 63-Series Guide Vane Profile NACA A4 K6 63-Series Guide Vane Profile (Continued) The general vane designation is 63 − (Cl0 A4 K6 )nn , where nn is the maximum thickness as percent of chord. The leading and trailing edge camberline slopes are infinite. An equivalent parabolic arc camberline can be used to provide viable definitions of leading and trailing edge blade angles. tan θ = O. Tuncer (ITU) 291.5cl0 468.75 − (5.83Cl0 )2 Axial Compressor (2) November 2011 77 / 148 Axial-Flow Compressor Blade Profiles Controlled Diffusion Airfoils Controlled Diffusion Airfoils I I I I I I I Standart blade profiles are used extensively for axial compressors. These are well-understood, reliable and can yield excellent performance if properly applied. However, many investigators have explored alternatives offering better Mach number range and efficiency. This is possible by specifying Mach number distributions on the surface. A continuous acceleration along the suction surface and near the leading edge to avoid boundary layer separation or premature separation. The peak Mach number should not exceed 1.3 in order to avoid shock-wave induced separation. Carefully controlled deceleration along the suction surface from the peak Mach number to avoid turbulent boundary layer separation ahead of the trailing edge. A nearly constant Mach number distribution on the pressure surface. O. Tuncer (ITU) Axial Compressor November 2011 78 / 148 Axial-Flow Compressor Blade Profiles Controlled Diffusion Airfoils Controlled Diffusion Airfoils (Continued) O. Tuncer (ITU) Axial Compressor November 2011 79 / 148 Axial-Flow Compressor Blade Profiles Blade Throat Opening Blade Throat Opening Minimum distance between adjacent blades. Governs the onset of local flow choking within the blade passage. O. Tuncer (ITU) Axial Compressor November 2011 80 / 148 Axial-Flow Compressor Blade Profiles Blade Throat Opening Blade Throat Opening for NACA-65 Series Blades Stagger Angle Parameter 1.5 1.5 φ = γ(1 − 0.05Cl0 ) + 5Cl0 −2 Throat Opening to Pitch Ratio √ √ o/s = [1 − tb σ/c) cos φ] σ O. Tuncer (ITU) Axial Compressor November 2011 81 / 148 Axial-Flow Compressor Blade Profiles Blade Throat Opening Extended Throat Opening Correlation O. Tuncer (ITU) Axial Compressor November 2011 82 / 148 Axial-Flow Compressor Blade Profiles Blade Throat Opening Staggered Blade Geometry Once the camberline and profile coordinates are generated along the chord, the geometry of the staggered blade in the cascade is obtained by a simple rotation of coordinates to the stagger angle γ. The staggered blade inlet and discharge angles are given by, κ1 = χ1 + γ κ2 = γ − χ2 θ = κ1 − κ2 For circular arc or NACA 65 series equivalent circular arc approximation. χ1 = χ2 = θ 2 γ = (κ1 + κ2 )/2 O. Tuncer (ITU) Axial Compressor November 2011 83 / 148 2-D Blade-to-Blade Flow Two-Dimensional Blade-to-Blade Flow Through Cascades of Blades I Offers a very natural view of cascade fluid dynamics to make it easier for designers to develop an understanding of the basic flow processes involved. I Inviscid blade-to-blade flow analysis addresses the general problem of two dimensional flow on a stream surface in an annular cascade. I Two dimensional boundary layer analysis can be used to approximate viscous effects. I Ignores secondary flows. I Loses accuracy when significant flow separation is present. O. Tuncer (ITU) Axial Compressor November 2011 84 / 148 2-D Blade-to-Blade Flow The Blade-to-Blade Flow Problem The Blade-to-Blade Plane Flow O. Tuncer (ITU) Axial Compressor November 2011 85 / 148 2-D Blade-to-Blade Flow The Blade-to-Blade Flow Problem The Blade-to-Blade Plane Stream Sheet O. Tuncer (ITU) Axial Compressor November 2011 86 / 148 2-D Blade-to-Blade Flow Coordinate System and Velocity Components Coordinate System and Velocity Components Convenient to use a coordinate transformation into new coordinates (ξ, η) such that blade surfaces correspond to η = constant. Z m dm 0 cos β η = [θ − θ0 ]/[θ1 − θ0 ] ξ = O. Tuncer (ITU) Axial Compressor November 2011 87 / 148 2-D Blade-to-Blade Flow Coordinate System and Velocity Components Coordinate System and Velocity Components r ∂θ tan β = ∂m O. Tuncer (ITU) = tan β0 + [tan β1 − tan β0 ]η η Axial Compressor November 2011 88 / 148 2-D Blade-to-Blade Flow Coordinate System and Velocity Components Coordinate System and Velocity Components The following equations relate velocity components to the more usual Wm and θ components. Wξ = Wm cos β + Wθ sin β Wq = Wθ cos β − Wm sin β Wm = Wξ cos β − Wq sin β Wθ = Wq cos β + Wz sin β O. Tuncer (ITU) Axial Compressor November 2011 89 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Potential Flow in the Blade-to-Blade Plane Assumptions, I Steady-state I Inviscid I Adiabatic I Rothaply and entropy constant on the flow plane. Conservation of Mass " 2∆m ρbWq cos β − m,η−∆η ρbwq cos β # +2∆η[(SρbWm )m−∆m,η −(SρbW m,η+∆η where, S = r (θ1 − θ0 ) O. Tuncer (ITU) Axial Compressor November 2011 90 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Conservation of Mass (Continued) Taking the limit, Continuity ∂ ρbWq ∂(SρbWm + =0 ∂η cos β ∂m Irrotational flow in the stream surface requires that the component of absolute vorticity normal to the stream sheet to be zero. ~ ) = ~en .[∇x( ~ + r ω~eθ )] = 0 ~ C ~ W ~en .(∇x Stokes Theorem I ~ .d~r = W C O. Tuncer (ITU) Z ~ )]da = − ~ W [~en .(∇x A Z ~ ω~eθ )]da ~en .(∇xr A Axial Compressor November 2011 91 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Irrotationality (Continued) Applying to control volume, " 2 ∆m Wξ cos β − m,η−∆η Wξ cos β # m,η+∆η + 2 ∆η[(SWθ )m−∆m,η − (SWθ )m+∆m,η ] = 4∆η∆m S ∂r 2 ω r ∂r Taking the limit, Wξ ∂ ∂(SWθ ) + 2Sω sin φ = ∂η cos β ∂m where φ is the stream sheet angle w.r.t the axial direction. sin φ = O. Tuncer (ITU) ∂r ∂m Axial Compressor November 2011 92 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Stream Function A strem function Ψ is defined by, ṁ ∂Ψ 0 − ρb(Wθ − Wm tan β) ∂m ṁ Ψ = SρbWm ∂η ṁ is the stream sheet mass flow rate. Velocity components are given by, Wm = Wθ = O. Tuncer (ITU) ṁ ∂Ψ Sbρ ∂η ṁ tan β ∂Ψ ∂Ψ − bρ S ∂η ∂m Axial Compressor November 2011 93 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Irrotationality in Terms of Stream Function Introducing stream function into the vorticity equation = ∂ ṁ(1 + tan2 β) ∂Ψ ṁ tan β ∂Ψ − ∂η Sbρ ∂η bρ ∂m ∂ ṁ tan β ∂Ψ ṁS ∂Ψ − + 2Sω sin φ ∂m bρ ∂η bρ ∂m This equation can be simplified into, A ∂2Ψ ∂Ψ ∂2Ψ ∂Ψ ∂2Ψ + C +E = 2Sω sin φ − 2B +D ∂η 2 ∂η∂m ∂m2 ∂η ∂m where, O. Tuncer (ITU) Axial Compressor November 2011 94 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Irrotationality in Terms of Stream Function (Continued) Boundary Conditions On the blade surfaces, Ψ(m, 0) = 0 Ψ(m, 1) = 1 O. Tuncer (ITU) Axial Compressor November 2011 95 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Irrotationality in Terms of Stream Function (Continued) Boundary Conditions (Continued) Periodic boundary conditions on the sides, Ψ(m, η + 1) = Ψ(m, η) + 1 ρ(m, η + 1) = ρ(m, η) Wm (m, η + 1) = Wm (m, η) Wθ (m, η + 1) = Wθ (m, η) Uniform flow at the upstream and downstream boundaries (This requires Ψ vary linearly in the tangential direction. Another choice is to require constant flow angle. If the geometry of the side boundaries coincide with the local flow angles. ∂Ψ ∂Ψ = cos β =0 ∂ξ ∂m O. Tuncer (ITU) Axial Compressor November 2011 96 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Irrotationality in Terms of Stream Function (Continued) Boundary Conditions (Continued) Periodic boundary conditions on the sides, Ψ(m, η + 1) = Ψ(m, η) + 1 ρ(m, η + 1) = ρ(m, η) Wm (m, η + 1) = Wm (m, η) Wθ (m, η + 1) = Wθ (m, η) Uniform flow at the upstream and downstream boundaries (This requires Ψ vary linearly in the tangential direction. Another choice is to require constant flow angle. If the geometry of the side boundaries coincide with the local flow angles. ∂Ψ ∂Ψ = cos β =0 ∂ξ ∂m O. Tuncer (ITU) Axial Compressor November 2011 97 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Irrotationality in Terms of Stream Function (Continued) Kutta Condition Typically β will be assigned to vary uniformly along the side boundary from the upstream boundary flow angle to the blade leading edge angle, and analogously for the downstream boundary. Upstream and downstream flow angles depend upon each other. A prediction of the downstream flow angle for any upstream flow condition is required. Therefore an additional constraint is needed. [W (m, 0)]te = [W (m, 1)]te Iterative adjustment of the discharge flow angle is needed until the above condition is met. O. Tuncer (ITU) Axial Compressor November 2011 98 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Irrotationality in Terms of Stream Function (Continued) Finite Difference Approximations For interior points, ∂Ψ ∂m ∂Ψ ∂η ∂2Ψ ∂m2 ∂2Ψ ∂m∂η O. Tuncer (ITU) = = = Ψi+1,j − Ψi−1,j 2∆m Ψi,j+1 − Ψi,j−1 2∆η Ψi+1,j − 2Ψi,j + Ψi−1,j (∆m)2 = Axial Compressor November 2011 99 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Irrotationality in Terms of Stream Function (Continued) Finite Difference Approximations For points at the boundary, ∂Ψ ∂m ∂Ψ ∂η ∂Ψ ∂m ∂Ψ ∂η O. Tuncer (ITU) = = = = 4Ψi+1,j − 3Ψi,j − Ψi+2,j 2∆m 4Ψi,j+1 − 3Ψi,j − Ψi,j+2 2∆η 3Ψi,j − 4Ψi−1,j + Ψi−2,j 2∆m 3Ψi,j − 4Ψi,j−1 + Ψi,j−2 2∆η Axial Compressor November 2011 100 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Irrotationality in Terms of Stream Function (Continued) Algebraic Form of the PDE for Interior Points Ψi,j + Ãi,j Ψi−1,j + B̃i,j Ψi+1,j + C̃i,j Ψi,j−1 + D̃i,j Ψi,j+1 + Ẽi,j [Ψi+1,j+1 − Ψi+1,j−1 − Ψi−1,j+1 + Ψi−1,j−1 ] = Q̃i,j where, O. Tuncer (ITU) Axial Compressor November 2011 101 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Irrotationality in Terms of Stream Function (Continued) Coefficients Ãi,j = B̃i,j = C̃i,j = Ei,j Ci,j − 2∆m (∆m)2 − 2A 2Ci,j i,j + (∆m) 2 (∆η)2 Ci,j Ei,j + 2∆m (∆m)2 − 2A 2Ci,j i,j + (∆m) 2 (∆η)2 Ai,j Di,j − 2∆η (∆η)2 − 2A 2Ci,j i,j + (∆m) 2 (∆η)2 O. Tuncer (ITU) Axial Compressor Di,j Ai,j + 2∆η (∆η)2 − 2A 2Ci,j i,j + (∆m) 2 (∆η)2 Bi,j 2∆m∆η 2Ai,j 2Ci,j + (∆m) 2 (∆η)2 D̃i,j = Ẽi,j = Q̃i,j = − 2sω sin φ 2Ai,j (∆η)2 + 2Ci,j (∆m)2 November 2011 102 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Solution Procedure I A relaxation technique or a matrix method can be used for solution. I After each solution for the stream function density field must be updated. Rothalpy is constant on the stream sheet, 1 1 1 h = H 0 − W 2 = I + (r ω)2 − W 2 (3) 2 2 2 Since entropy is also constant, all thermodynamic properties can be calculated from (h,s). Density can be calculated using an appropriate equation of state. I As long as the flow is subsonic this lagging density solution offers very good numerical stability and rapid convergence. I If the flow is transonic or supersonic other solution procedures are needed as the PDE is no longer elliptic in nature. I Stability of numerical solution is significantly influenced by the grid structure. O. Tuncer (ITU) Axial Compressor November 2011 103 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Leading Edge Grid Structure O. Tuncer (ITU) Axial Compressor November 2011 104 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Potential Flow Results O. Tuncer (ITU) Axial Compressor November 2011 105 / 148 2-D Blade-to-Blade Flow Potential Flow in the Blade-to-Blade Plane Transonic Potential Flow Results O. Tuncer (ITU) Axial Compressor November 2011 106 / 148 2-D Blade-to-Blade Flow Linearized Potential Flow Analysis Linearized Potential Flow Analysis Motivation The real purpose of the linearized method is its use in a quasi-three-dimensional flow analysis, where blade to blade flow analyses are conducted on several stream sheets and must be repeated many times. Linearization of the Stream Function Ψ(m, n) = a(m)[η − η 2 ] + η 2 This is tantamount to assuming ρbWm vary linearly with η. Note that W = Wξ on the blade surface. W1 W0 − = β1 cos β0 O. Tuncer (ITU) Z 0 1 ∂SWθ + 2Sω sin φ ∂m Axial Compressor November 2011 107 / 148 2-D Blade-to-Blade Flow Linearized Potential Flow Analysis Linearized Potential Flow Analysis (Continued) The velocity normal to the blade surfaces must be zero, W = Wm 1 ṁ ∂Ψ = cos β cos β Sbρ ∂η Combining, W1 cos β1 W0 cos β0 = = ṁ(2 − a) Sbρ cos2 β1 ṁa Sbρ cos2 β0 SWθ = ṁ[tan β(a − 2aη + 2η) − a0 S(η − η 2 )]/(bρ) where the prime denotes total derivative w.r.t. m. O. Tuncer (ITU) Axial Compressor November 2011 108 / 148 2-D Blade-to-Blade Flow Linearized Potential Flow Analysis Linearized Potential Flow Analysis (Continued) In order to simplify the analysis we define, u(m, η) = ṁ tan β/(bρ) v (m, η) = ṁS/(bρ) Differentiating the previous eqaution and substituting new variables, ∂SWθ ∂u ∂v 0 = (a − 2aη − 2η) + (1 − 2η)ua0 − (va00 + a )(η − η 2 ) ∂m ∂m ∂m O. Tuncer (ITU) Axial Compressor November 2011 109 / 148 2-D Blade-to-Blade Flow Linearized Potential Flow Analysis Linearized Potential Flow Analysis (Continued) Using truncated Taylor series expansion for any function F (η), for values at 0, 0.5, and 1, a three-point difference approximation to the integral is obtained. Z 1 F (η)dη = (F0 + 4F̄ + F1 )/6 0 where overbar denotes the function value at η = 0.5. Therefore, Z 0 1 ∂SWθ dη = [au00 + u0 a0 + 4ū 0 − v̄ a00 − v̄ 0 a0 + u10 (2 − a) − u1 a0 ]/6 ∂m Combining, a00 + Aa0 + Ba = C where A,B, and C are functions of m only. O. Tuncer (ITU) Axial Compressor November 2011 110 / 148 2-D Blade-to-Blade Flow Linearized Potential Flow Analysis Linearized Potential Flow Analysis (Continued) A(m) = [v̄ 0 − u0 + u1 ]/v̄ B(m) = u10 − u00 v0 6 v1 + − v̄ v̄ S 2 cos2 β1 cos2 β0 C (m) = 2u10 + 4ū 0 + 12ω sin φ 12v1 − 2 v̄ v̄ S cos2 β1 O. Tuncer (ITU) Axial Compressor November 2011 111 / 148 2-D Blade-to-Blade Flow Linearized Potential Flow Analysis Linearized Potential Flow Analysis (Continued) Leading Edge Boundary Condition The leading edge boundary condition follows from the known inlet angular momentum supplied by the upstream flow, Wθ,in . Integrating Wθ across the passage at the leading edge, a0 + a[u1 − u0 ]/v̄ = Kutta Condition Kutta condition is used as the trailing edge boundary condition (i.e. W0 = W1 ). a = 2cosβ0 /[cos β1 + cos β0 ] O. Tuncer (ITU) Axial Compressor November 2011 112 / 148 2-D Blade-to-Blade Flow Linearized Potential Flow Analysis Solution Method I The governing equations are cast into matrix form, Ax = B. I Matrix A is in tri-diagonal form. Except for the equation at the leading edge point. I Gas density field update is lagging the velocity field. Same procedure as before. O. Tuncer (ITU) Axial Compressor November 2011 113 / 148 2-D Blade-to-Blade Flow Linearized Potential Flow Analysis Tri-Diagonal Systems Tri-diagonal systems commonly arise in the solution of many engineering problems. Ordinary matrix inversion for a tri-diagonal system with n unknowns requires O(n3 ) operations. Such an approach is unnecessarily computationally intensive. “Thomas Algorithm”only requires O(n) operations. A tri-diagonal system can be written as, ai xi−1 + bi xi + ci xi+1 = di , with, a1 = 0, cn = 0 This system can be cast into matrix Ax = b form with, b1 c1 0 · · · 0 0 x1 a2 b2 c2 · · · 0 0 x2 b3 c3 0 A = 0 a3 , x = .. , b = . .. .. 0 ... . . 0 xn 0 0 ··· an bn d1 d2 .. . dn Note that in matrix A all entries are zero except the ones in the diagonal, the super-diagonal and the sub-diagonal. O. Tuncer (ITU) Axial Compressor November 2011 114 / 148 2-D Blade-to-Blade Flow Linearized Potential Flow Analysis Thomas Algorithm The solution of this system is performed in two steps as per the Thomas Algorithm. First step involves modifying the coefficient vectors. (c 1 i =1 ci0 = b1 ci i = 2, 3, · · · , n − 1 0 a bi −ci−1 i d1 i =1 b 0 a di0 = d1i −di−1 i b −c 0 a i = 2, 3, · · · , n − 1 i i i−1 After the new coefficients are obtained solution is reached through back substitution. ( i =n dn0 xi = 0 0 di − ci xi+1 i = n − 1, n − 2, · · · , 1 O. Tuncer (ITU) Axial Compressor November 2011 115 / 148 2-D Blade-to-Blade Flow Linearized Potential Flow Analysis Listing 1: Source Code to Implement the Thomas Algorithm i n t Thomas ( const double ∗a , const double ∗b , double ∗ c double ∗d , double ∗x , i n t n ) { int i ; /∗ Mo di fy t h e c o e f f i c i e n t s . ∗/ c [ 0 ] /= b [ 0 ] ; d [ 0 ] /= b [ 0 ] ; f o r ( i = 1 ; i < n ; i ++){ double i d = 1 . 0 / ( b [ i ] − c [ i − 1 ] ∗ a [ i ] ) c [ i ] ∗= i d ; /∗ L a s t v a l u e i s r e d u n d a n t d [ i ] = (d [ i ] − d [ i − 1]∗ a [ i ])∗ id ; } /∗ Back s u b s t i t u t i o n ∗/ x [ n − 1] = d [ n − 1 ] ; f o r ( i = n − 2 ; i >= 0 ; i −−) x [ i ] = d [ i ] − c [ i ]∗ x [ i + 1 ] ; return 0; } O. Tuncer (ITU) Axial Compressor November 2011 116 / 148 2-D Blade-to-Blade Flow Linearized Potential Flow Analysis Linearized Potential Flow Results O. Tuncer (ITU) Axial Compressor November 2011 117 / 148 2-D Blade-to-Blade Flow Time Marching Method Time Marching Method I Potential flow method has limitations in terms of Mach number. I Time marching method provides a more general solution capability. I It is applicable to subsonic, transonic and supersonic flows. I Suggested first by von Neumann and Richtmayer (1950). I Solution is accomplished using the same velocity components and coordinate system. I The governing equations are solved in their full unsteady form. I Solution is advanced in time until variations is time become negligible. O. Tuncer (ITU) Axial Compressor November 2011 118 / 148 2-D Blade-to-Blade Flow Time Marching Method Time Marching Method (Integral form of Conservation Equations) Mass Z V ∂ρ dV + ∂t Z ~ .~n)dA = 0 ρ(W A Momentum Z V ~ ∂ρW dV + ∂t Z ~ (W ~ .~n)dA + ρW Z A Z P~e (~e .~n)dA = A ~f dV V Energy Z Z Z ∂H 0 ∂P ~ .~n)dA = ~ )dV ρ − dV + ρH 0 (W ρ(~f .W ∂t ∂t V A V O. Tuncer (ITU) Axial Compressor November 2011 119 / 148 2-D Blade-to-Blade Flow Time Marching Method Time Marching Method (Integral form of Conservation Equations) ~e is the unit vector along w ~. ~f is a body force used to account for the Coriolis and centrifugal acceleration terms in the rotating curvilinear system. Equations in conservative form, Sb Sb ∂ρWm ∂t + = O. Tuncer (ITU) ∂ρ ∂ ∂ + [SbρWm ] + [bρQ] = 0 ∂t ∂m ∂η ∂ ∂bP ∂ [Sb(ρWm2 + P)] − tan β + [bρQWm ] ∂m ∂η ∂η 1 ∂Sb SBρ sin φ(Wθ + r ω)2 + P r ∂m Axial Compressor November 2011 120 / 148 2-D Blade-to-Blade Flow Time Marching Method Time Marching Method (Integral form of Conservation Equations) Sb ∂ρWθ 1 ∂ ∂ ∂ + [Sb(ρWm (Wθ +r ω)]+ [b(QWθ +P)] = r ω [SbρWm ] ∂t r ∂m ∂η ∂m Sb ∂(ρI − P) ∂ ∂ + [SbρWm I ] + [bρQI = 0 ∂t ∂m ∂η Q is a special velocity component to conserve properties at the constant η boundaries of the control cell. Q = Wq / cos β = Wθ − Wm tan β O. Tuncer (ITU) Axial Compressor November 2011 121 / 148 2-D Blade-to-Blade Flow Time Marching Method Time Marching Method (Integral form of Conservation Equations) Note that Q = Wq = 0 on the blade surface. For these points there is only Wξ component of velocity. Applying the integral momentum equation in the ξ-direction. Sb ∂ρWξ ∂t ∂ ∂ [Sb(ρWm Wξ + P cos β)] + [bρQWξ ] ∂m ∂η ∂ [Sb cos β] + Sbρ sin φ cos βr ω 2 = P ∂m + Wm = Wξ cos β Wθ = Wξ sin β O. Tuncer (ITU) Axial Compressor November 2011 122 / 148 2-D Blade-to-Blade Flow Time Marching Method Time Marching Method (Boundary Conditions) I On the blades velocity normal to the surface is zero. I For side boundaries outside the blade passage periodic boundary conditions are used (as before) I Upstream and downstream boundary conditions are rather tricky. I O. Tuncer (ITU) Axial Compressor November 2011 123 / 148 2-D Blade-to-Blade Flow Time Marching Method Unsteady Characterisctics dm dt dm dt dm dt O. Tuncer (ITU) = Wm + a = Wm − a = Wm Axial Compressor November 2011 124 / 148 2-D Blade-to-Blade Flow Time Marching Method Subsonic Upstream Boundary At t + ∆t one of the characteristics is inside the domain. A logical choice for the computed dependant variable is density. P, Wm from equation of state and rothalpy (if entropy is known). At the upstream boundary specify Wθ , Pt , Tt . O. Tuncer (ITU) Axial Compressor November 2011 125 / 148 2-D Blade-to-Blade Flow Time Marching Method Supersonic Upstream Boundary All characteristics outside the solution domain. All dependant variables must be assigned when Wm > a on an upstream boundary. O. Tuncer (ITU) Axial Compressor November 2011 126 / 148 2-D Blade-to-Blade Flow Time Marching Method Subsonic Downstream Boundary At t + ∆t one of the characteristics is inside the domain. Specify discharge static static pressure as the boundary condition. O. Tuncer (ITU) Axial Compressor November 2011 127 / 148 2-D Blade-to-Blade Flow Time Marching Method Supersonic Downstream Boundary All characteristics inside the solution domain. All dependent variables can be computed from the solution without any boundary condition specification. O. Tuncer (ITU) Axial Compressor November 2011 128 / 148 2-D Blade-to-Blade Flow Time Marching Method Numerical Stability I No Kutta condition is needed! I Specifiying more boundary conditions WILL NOT PRODUCE A VALID SOLUTION. WILL CAUSE THE SOLUTION TO DIVERGE!!! I Explicit solution scheme numerically unstable. I Stabilizing terms (like artificial viscosity) is needed in governing equations if an explicit solution is sought. O. Tuncer (ITU) Axial Compressor November 2011 129 / 148 2-D Blade-to-Blade Flow Time Marching Method Stability Analysis ut = v (ξ, η, t) + µ(ξ) uξξ + µ(η) uηη For a stable solution the coefficients must satisfy the following conditions. µ(ξ) ≥ µ(η) ≥ µ(ξ) ≥ µ(η) ≥ O. Tuncer (ITU) 1 (|Wm | + a)2 ∆t 2 1 (|Wθ | + a)2 ∆t 2 1 [(|Wξ | + a) cos β]2 ∆t 2 1 [(|Wq | + a)/ cos β]2 ∆t 2 Axial Compressor November 2011 130 / 148 2-D Blade-to-Blade Flow Time Marching Method Stability Analysis (Continued) If the grid structure is highly skewed and node spacing in the tangential direction is much finer than the meridional direction an additional meridional stabilizing term can be added. 1 µ(ξ) → µ(ξ) [(|Wθ | + a) sin2 β(∆m)/(S∆η)]2 ∆t 2 O. Tuncer (ITU) Axial Compressor November 2011 131 / 148 2-D Blade-to-Blade Flow Time Marching Method Stability Analysis (Continued) CFL Criterion ∆tmax ≤ ∆tmax ≤ ∆m |Wm | + a ∆S∆η |Wθ | + a Actual time step is, ∆t = µ0 ∆tmax A rule-of-thumb is, 0.1 ≤ µ0 ≤ 0.9 O. Tuncer (ITU) Axial Compressor November 2011 132 / 148 2-D Blade-to-Blade Flow Time Marching Method Stability Analysis (Continued) If the time derivative is approximated by a forward time derivative, u(ξ, η, t + ∆t) = u(ξ, η, t) + [v (ξ, η, t) + µ(ξ) uξξ + µη uηη ]∆t I Stabilizing terms second order w.r.t ∆t I Dynamic terms of the original PDE are first order w.r.t ∆t I Start with a large time step a the beginning I Reduce the time step as the solution converges O. Tuncer (ITU) Axial Compressor November 2011 133 / 148 2-D Blade-to-Blade Flow Time Marching Method Time Marching Solution Results O. Tuncer (ITU) Axial Compressor November 2011 134 / 148 2-D Blade-to-Blade Flow Time Marching Method Transonic Time Marching Solution Results O. Tuncer (ITU) Axial Compressor November 2011 135 / 148 2-D Blade-to-Blade Flow Blade Surface Boundary Layer Analysis Blade Surface Boundary Layer Analysis Motivation I Quantify viscous effects I Predict flow seperation I Predict the level of total pressure loss Momentum Integral Equation 1 ∂bρe ue2 θ ∂ue + δ ∗ ρe ue = τw b ∂x ∂x O. Tuncer (ITU) Axial Compressor November 2011 136 / 148 2-D Blade-to-Blade Flow Blade Surface Boundary Layer Analysis Boundary Layer Analysis (Continued) Momentum Thickness ρe We2 θ Z ρW (We − W )dy = Displacement Thickness ∗ Z δ (ρe We − ρW )dy ρe We δ = 0 Skin Friction Coefficient Cf = O. Tuncer (ITU) τw 1 2 2 ρe We Axial Compressor November 2011 137 / 148 2-D Blade-to-Blade Flow Blade Surface Boundary Layer Analysis Boundary Layer Analysis of Grushwithz (1950) Boundary layer initially laminar, soon transitions into turbulence. Universal Boundary Layer Profile W = C1 η + C2 η 2 + C3 η 3 + C4 η 4 We Z 1 y ρ δ 0 0 ρe Z δ ρ = 0 ρe η = δ0 O. Tuncer (ITU) Axial Compressor November 2011 138 / 148 2-D Blade-to-Blade Flow Blade Surface Boundary Layer Analysis Boundary Layer Analysis of Grushwithz (1950) Shape Factor Λ= ρ2e (δ 0 )2 dWe ρw µ dx Matching Edge Conditions C1 = 2 + Λ/6 C2 = −Λ/2 C3 = Λ/2 − 2 C4 = 1 − Λ/6 O. Tuncer (ITU) Axial Compressor November 2011 139 / 148 2-D Blade-to-Blade Flow Blade Surface Boundary Layer Analysis Boundary Layer Analysis of Grushwithz (1950) Momentum Thickness Λ Λ2 θ 37 − − = δ0 315 945 9072 Energy Thickness δE = δ0 Z 0 δ ρW W2 798048 − 4656Λ − 758Λ2 − 7Λ3 1 − 2 dy = ρe We We 4324320 Velocity Thickness δW = δ0 O. Tuncer (ITU) Z 0 δ ρW W 3 Λ FWe2 − + 1− dy = ρ e We We 10 120 2cp Te Axial Compressor November 2011 140 / 148 2-D Blade-to-Blade Flow Blade Surface Boundary Layer Analysis Boundary Layer Analysis of Grushwithz (1950) Velocity Thickness Λ 3 Λ Λ 2 + 17.8063 F = 0.232912 − 0.831483 + 0.650584 100 100 100 Enthalpy Thickness For adiabatic walls with Pr = 1 δh = δ0 Z 0 δ ρW ρe We h − 1 dy = he Displacement Thickness δ ∗ = δh + δw O. Tuncer (ITU) Axial Compressor November 2011 141 / 148 2-D Blade-to-Blade Flow Blade Surface Boundary Layer Analysis Boundary Layer Analysis of Grushwithz (1950) Following parameters introduced for convenience, b0 = ρe T0 = t ρ Te 2 θ ρe θ2 dWe K = Λ 0 = b0 δ µ dx Gruschweitz (1950) shows that, 1 µ Λ cf = 1 + 2 ρe We δ 0 6 37 Λ Λ2 K= − − 315 945 9072 O. Tuncer (ITU) Axial Compressor 2 Λ November 2011 142 / 148 2-D Blade-to-Blade Flow Blade Surface Boundary Layer Analysis Boundary Layer Analysis of Grushwithz (1950) Transition Criteria <θ = ρ e We θ > 250 µ Once boundary layer transitions into turbulence different methods are needed. Method of Head (1958) Originally developed for incompressible boundary layers. ∂ [bρe We (δ − δ ∗ )] = bρe We E ∂x Empirical relations are needed for E and cf as a function of θ and (δ − δ ∗ ). O. Tuncer (ITU) Axial Compressor November 2011 143 / 148 2-D Blade-to-Blade Flow Blade Surface Boundary Layer Analysis Boundary Layer Analysis of Grushwithz (1950) Shape Factors H1 ≡ (δ − δ ∗ )/θ H = δ ∗ /θ Kinematic Shape Factor Recommended by Green (1968) 1 Hk = θ δ Z 0 ρ ρe W 1− We dy For adiabatic walls with Pr = 1 H = (Hk + 1)Tt0 /Te − 1 O. Tuncer (ITU) Axial Compressor November 2011 144 / 148 2-D Blade-to-Blade Flow Blade Surface Boundary Layer Analysis Boundary Layer Analysis of Grushwithz (1950) Aungier 2003 recommends, Hk = 1 + [0.9/(H1 − 3.3)]0.75 E = 0.025(Hk − 1) Skin friction coefficient correlation of Ludwieg and Tillmann (1950) commonly used for incompressible turbulent boundary layer analysis. cf ,inc = 0.246exp(−1.561Hk )Reθ−0.268 Green’s correction to it, cf = cf ,inc O. Tuncer (ITU) Axial Compressor November 2011 145 / 148 2-D Blade-to-Blade Flow Blade Surface Boundary Layer Analysis Boundary Layer Analysis of Grushwithz (1950) At the transition (laminar to turbulent) point, (δ − δ ∗ )turb = (δ − δ ∗ )lam θturb = θlam From Gruschwitz (1950) profiles, δ − δ∗ = δ0 7 Λ + 10 120 Seperation Criterion for Turbulent Boundary Layers Hk ≥ 2.4 O. Tuncer (ITU) Axial Compressor November 2011 146 / 148 2-D Blade-to-Blade Flow Blade Surface Boundary Layer Analysis Boundary Layer Analysis of Grushwithz (1950) Useful to predict total pressure loss coefficient for the cascade from the B.L. data at the trailing edge. Total pressure loss coefficient based on cascade inlet velocity (Lieblein and Roudebush, 1956), where the summation is carried out for the boundary layers on both blade surfaces. This loss coefficient can be used to estimate the rotor efficiency. cos βin 2 2Θ + (∆∗ )2 cos βout (1 − ∆∗ )2 P θ Θ= S cos βout P ∗ δ ∗ ∆ = S cos βout ∆Pt ω̄ = = (Pt − P)in Loss coefficients are approximate, since the analysis method ignores secondary flows. O. Tuncer (ITU) Axial Compressor November 2011 147 / 148 2-D Blade-to-Blade Flow Blade Surface Boundary Layer Analysis Boundary Layer Analysis Results O. Tuncer (ITU) Axial Compressor November 2011 148 / 148