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Part III: the Viscous Flow

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Part III: the Viscous Flow Why does an airfoil drag: the viscous problem – André Deperrois – March 2019 Rev. 1.1 © Navier-Stokes equations Inviscid fluid Time averaged turbulence CFD « RANS » Reynolds Averaged Euler’s equations Reynolds equations Navier-stokes solvers irrotational flow Viscosity models, uniform 3d Boundary Layer eq. pressure in BL thickness, Prandlt Potential flow mixing length hypothesis. 2d BL equations Time independent, incompressible flow Laplace’s equation 1d BL Integral 2d BL differential equations equations 2d mixed empirical + theoretical 2d, 3d turbulence and transition models 2d viscous results interpolation The inviscid flow around an airfoil Favourable pressure gradient, the flo! a""elerates ro# $ero at the leading edge%s stagnation point& Adverse pressure gradient, the low decelerates way from the surface, the flow free tends asymptotically towards the stream air freestream uniform flow flow inviscid ◀—▶ “laminar”, The boundary layer way from the surface, the fluid’s velocity tends !ue to viscosity, the asymptotically towards the tangential velocity at the velocity field of an ideal inviscid contact of the foil is " free flow around an airfoil$ stream air flow (magnified scale) The boundary layer is defined as the flow between the foil’s surface and the thic%ness where the fluid#s velocity reaches &&' or &&$(' of the inviscid flow’s velocity. The viscous flow around an airfoil at low Reynolds number Favourable pressure gradient, the low a""elerates ro# $ero at the leading edge%s stagnation point& Adverse pressure gradient, the low decelerates +n adverse pressure gradients, the laminar separation bubble forms. The flow goes flow separates. The velocity close to the progressively turbulent inside the bubble$ surface goes negative. The turbulent flow reattaches, i.e. the velocity reverts to positive throughout the ,* *aminar flow )tagnation point where the flow separates and the tangential The turbulent flow separates velocity is " The viscous flow around an airfoil y 'op surface s laminar flow surf. velocity lam. separation turbulent turbulent wa%e decreases reattachment separation s 'hings to ● The ,* thic%ness increases progressively note .o surface slip on the airfoil surface The viscous flow around an airfoil separation bubble forms$ The /avorable pressure gradient, the dverse pressure gradient, the flow flow goes progressively turbulent flow accelerates$ decelerates and separates$ inside the bubble$ (L edge velocity free calculated by *Foil The turbulent flow stream air reattaches, i.e. the flow surface velocity reverts to positive Stagnation point where the flow divides and the tangential velocity is " The viscous flow around an airfoil The transition problem The transition from laminar to turbulent flow is a complex problem in 1d and even more so in 2d$ free Things to note in 1d: stream air ● Transition occurs when the amplification factor of spatial waves %nown as flow Tollmien–Schlichting waves reaches a critical value, i$e$ the .4rit factor ● Turbulent flow starts with small “sparks” which eventually extend downstream to full turbulence ● “/or low Reynolds number flows, the transition is separation induced” 5in.T. 4ebeci, 6odeling and computation of boundary-layer flows, chapter.($18 This is also what 9foil predicts The 2d problem The 2d inviscid potential problem can be The velocity field is used as an solved numerically for the velocity field by input to solve the ,* problem solving Laplace’s equation The 2d problem The 2d inviscid potential problem can be The velocity field is used as an solved numerically for the velocity field input to solve the ,* problem +n the ;940s, theoreticians have found that this method does not converge in adverse pressure gradients, e.g. on the upper surface of an airfoil$ This problem is %nown as the “=oldstein singularity” The 2d problem The 2d inviscid potential problem can be The velocity field is used as an solved numerically for the velocity field input to solve the ,* problem The reason is that the ,* disturbs the inviscid flow: the foil behaves as if it had an additional thic%ness The invis"id flo! !ants to dictate This additional thic%ness is called the its la! to the (), but the () does “displacement thic%ness δ>” not agree, and vi"e versa& .ote: not the same thing as the ,* thic%ness The 2d problem The 2d inviscid potential problem can be The velocity field is used as an solved numerically for the velocity field input to solve the ,* problem The reason is that the ,* disturbs the The inviscid inviscid flow: the foil behaves as if it had an flow needs to additional thic%ness be updated This iterative method is called the “Interactive Boundary Layer”, or ,() The 2d problem The inviscid velocity is used as “!irect” : an input for the ,* solver The viscous velocity from the ,* Many schemes have been proposed for “+nverse” solution is used as an input for the +,L problem the potential flow solver “)imultaneous” : The inviscid and ,L e:uations are solved concurrently at each iteration Developed by .& Drela and /& 0oungren in the 1223s, *Foil is still a 4the56 state-of-the-art ,(L solver some 73 years later About XFoil A co#prehensive set of =d (L turbulen"e and transition #odels 'hree #ain things !hi"h #ake *Foil outstanding A ull si#ultaneous ,(L solver A robust and reliable so tware package http:99!eb&#it.edu/drela9:ubli"/papers9; oil_sv.pd About XFoil *Foil’s 1D Integral method () equations are integrated in the () thickness () properties are therefore function only of the stream!ise position “s- 3 variables, one space di#ension )a#inar lo!8 the ,* properties are 'urbulent lo!8 the ,* properties are defined by defined by 7 the displacement thic%ness δ> - the displacement thic%ness δ> 7 the momentum thic%ness θ - the momentum thic%ness θ 7 the amplification factor n - the ma0. shear stress coeff$ 4τ s http:99!eb&#it.edu/drela9:ubli"/papers9; oil_sv.pd Standard behaviour of About XFoil a () at low Re ?rogressive transition from laminar to Turbulent Turbulent wa%e turbulent inside the reattachment separation bubble Turbulent *aminar separation separation 5not always8 !isplacement Momentum thic%ness δ> thickness θ !ifferential solvers ?ith the increase of "omputing power, it has become possible to solve the (L equations !ithout prior integration in the thi"kness& (L properties are de ined at each position 4s, y6& y s Although they require less e#piri"al assu#ptions than integral #ethods, di erential solvers still need a turbulence #odel !hich is the key building brick of the #ethod& !ifferential solvers The results which follow are from a differential solver currently in development$ ● The solver is based on the methods proposed by T. Cebeci in “Modeling and computation of boundary-layer flows” ● +t uses the Cebeci-Smith turbulence model s Differential solver *aminar attached flow *aminar separation Turbulent separated Wa%e flow ,* thic%ness @ "$""< ,* thickness @ "$";( ,* thickness @ "$"1" ,* thic%ness @ "$"A Clark 0 air oil, 'op surface, Re@=33k, aoa=3°, unit chord length BL thickness vs. Displacement thickness δ δ* XFoil vs. a differential solver )o why does an airfoil dragB 'he viscosity creates drag forces by t!o e e"ts ➔ it creates skin fri"tion orces on + riction drag- the air oil’s sur ace ➔ it creates unbalanced pressure +pressure drag- orces on the air oil’s sur ace s Note8 The induced drag is a 2d effect only, and is pressure forces not related to viscosity skin friction forces )o why does an airfoil dragB @3 in inviscid flo!8 'he viscosity creates drag forces by ”no viscosity, no t!o e e"ts ri"tion- ➔ it creates skin fri"tion orces on + riction drag- the air oil’s sur ace ➔ it creates unbalanced pressure +pressure drag- @3 in inviscid flo! orces on the air oil’s sur ace due to d%Alembert’s parado; s Note8 The induced drag is a 2d effect only, and is pressure forces not related to viscosity skin friction forces )o why does an airfoil dragB + ri"tion drag- + “pressure drag” @ “Viscous drag” or “Pro ile drag- (oth ter#s are used inter"hangeably in x lrD s pressure forces skin friction forces )o why does an airfoil dragB + ri"tion drag- + “pressure drag” @ “Viscous drag” or “Pro ile drag- Note8 The direct evaluation of friction and pressure forces is numerically unreliableC X/oil#s method is to evaluate the total viscous drag in the wake using the )quire7Doung formula s pressure forces skin friction forces -That%s it- In the hope that the concepts, wording, graphs, limitations and possibilities of 9Foil and xflr5 are a little more clear now than they were at the start of these presentations$ – André Deperrois – March 2019 Rev. 1.1 ©.
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