Introduction – Equations of motion G. Dimitriadis 03

Aeroelasticity Lecture 3: Unsteady Aerodynamics – Theodorsen G Dimitriadis Introduction to Aeroelasticity Unsteady Aerodynamics •! As mentioned in the first lecture, quasi-steady aerodynamics ignores the effect of the wake on the flow around the airfoil •! The effect of the wake can be quite significant •! It effectively reduces the magnitude of the aerodynamic forces acting on the airfoil •! This reduction can have a significant effect on the values of the flutter Introduction to Aeroelasticity 2D wing oscillations •! Consider a 2D airfoil oscillating sinusoidally in an airflow •! The oscillations will result in changes in the circulation around the airfoil •! Kelvin’s theorem states that the change in circulation over the entire flowfield must always be zero •! Therefore, any increase in the circulation around the airfoil must result in a decrease in the circulation of the wake •! In other words, the wake contains a significant amount of circulation, which balances the changes in circulation over the airfoil •! It follows that the wake cannot be ignored in the calculation of the forces acting on the airfoil Introduction to Aeroelasticity Kelvin’s Theorem •! The theorem states that: !" =0 !t •! For the oscillating airfoil problem, this means that: !airfoil ( t ) + !wake ( t ) = !0 •! Where !0 is the total circulation at time t=0 Introduction to Aeroelasticity Pitching and Heaving Wake shape of a sinusoidally pitching and heaving airfoil Positive vorticity is denoted by red and negative by blue Introduction to Aeroelasticity Experimental results Wake vorticity is a realworld phenomenon Here is a comparison between numerical simulation results (top) and flow visualization in a water tunnel (bottom) by Jones and Platzer Introduction to Aeroelasticity How to model this? •! The simulation results are useful but –! Not always accurate (there can be problems concerning starting vortices for example) –! Not practical If the motion (or any of the parameters) is changed, a new simulation must be performed •! Analytical mathematical models of the problem exist They were developed in the 1920s and 1930s •! Most popular models: –! Theodorsen –! Wagner Introduction to Aeroelasticity Simplifications •! In Theodorsen’s approach, only three major simplifications are assumed: –! The flow is always attached, i.e the motion’s amplitude is small –! The wing is a flat plate –! The wake is flat •! The flat plate assumption is not problematic In fact Theodorsen worked on a flat plate with a control surface (3 d.o.f.s), so asymmetric wings can also be handled •! If the motion is small (first assumption) then the flat wake assumption has little influence on the results Introduction to Aeroelasticity Basis of the model •! The model is based on elementary solutions of the Laplace equation: ! 2" = •! Such solutions are: –! The free stream: –! The source and the sink: –! The vortex: –! The doublet: Introduction to Aeroelasticity ! = U cos"x + U sin "y " " 2 ln r = ln ( x $ x ) + ( y $ y ) 2# 2# & y " y0 ) # # ! = " % = " tan "1 ( + 2$ 2$ ' x " x0 * µ cos # µ x != = 2" r 2" x + y != Circle •! Theodorsen chose to model the wing as a circle that can be mapped onto a flat plate through a conformal transformation: Introduction to Aeroelasticity Discussion •! Theodorsen’s approach has led to equations (5) and (6) for the full lift and moment acting on the airfoil •! The main assumptions of the approach are: –! Attached flow everywhere –! The wake is flat –! The wake vorticity travels at the free stream airspeed •! In all other aspects it’s an exact solution •! However, it’s not complete yet What is the value of C? Introduction to Aeroelasticity Prescribed motion •! In order to carry out the integrals and define C we need to know V •! The only way to know V is the prescribe it •! However, prescribing V directly is not useful •! It’s better to prescribe the wing’s motion and then determine what the resulting value of V will be Introduction to Aeroelasticity Sinusoidal motion •! The most logical choice for prescribed motion is sinusoidal motion Slowly pitching and plunging airfoil Vorticity variation with x/c in the wake It is sinusoidal near the airfoil Introduction to Aeroelasticity More about sinusoidal motion •! For small amplitude and frequency oscillations, the form of V(x0,t) is sinusoidal near the airfoil •! Notice that V(x0,t) is periodic in both time and space: –! V(x0,t)=V(x0,t+2#/$) = –! V(x0,t)=V(x0+U2#/$,t) = •! This means that phase angle of V(x0,t) is given by $t+$x0/U Introduction to Aeroelasticity Vortex strength V(x0,t) !(t) x0 h(t) The vortex strength of the wake behind a pitching and plunging airfoil can have any spatial and temporal distribution, V(x0,t) There are two special motions for which Theodorsen’s function can be evaluated: steady motion and sinusoidal motion For sinusoidal motion: Introduction to Aeroelasticity ! = ! 0e j"t h = h0e j"t V = V0e " & # j % "t + x ( $ U ' = V0e b" & # j % "t + x ( $ U 0' Theodorsen Function For sinusoidal motion Theodorsen’s function can be evaluated in terms of Bessel functions of the first and second kind A much more practical, approximate, estimation is: With k="c/U or k="b/U Introduction to Aeroelasticity Usage of Theodorsen •! Theodorsen’s lift force is now given by % % 3c ' ˙' ˙ $ x f #* lc = !"UcC ( k )) U# + h + & ( ( & •! Theodorsen’s function can be seen as an analog filter It attenuates the lift force by an amount that depends on the frequency of oscillation •! Theoretically, Theodorsen’s function can only be applied in the case where the response of the system is exactly sinusoidal Introduction to Aeroelasticity Example Consider the circulatory lift of a purely pitching flat plate, h=h0exp j$t Quasi-steady lift: lc = !"Uch0 j# exp j#t Theodorsen lift: lc = !"UcC ( k ) h0 j# exp j#t Introduction to Aeroelasticity Lift and moment •! The full equations for the lift and moment around the flexural axis using Theodorsen are: Introduction to Aeroelasticity Aeroelastic equations •! The full aeroelastic equations are: " m S % ( h˙˙+ " K h $ ') , + $ # S I! & *!˙˙ - # 0 % ( h + (.l( t )+ , ') , = ) K! & *! - * m( t ) - •! For sinusoidal motion they become: $ !" m + K h & % !" S !" S ' * h0 - j"t *!l( t ) ) +# e = + !" I# + K# ( , / , m(t ) / •! Substituting for l(t) and m(t) yields … Introduction to Aeroelasticity Equations of motion? •! As the system is assumed to respond sinusoidaly there is no sense in writing out complete equations of motion •! Combining the lift and moment with the structural forces gives Introduction to Aeroelasticity Validity of this equation •! This algebraic system of equations is only valid when the wing is performing sinusoidal oscillations •! For an aeroelastic system such oscillations are only possible when: –! The airspeed is zero and there is no structural damping – free sinusoidal oscillations –! There is an external sinusoidal excitation force – forced sinusoidal oscillations –! The wing is flying at the critical flutter condition – selfexcited sinusoidal oscillations •! The last case is very useful for calculating the critical flutter condition Introduction to Aeroelasticity Flutter Determinant •! For the equation to be satisfied non-trivially, the 2x2 matrix must be equal to zero, i.e D=0, where •! D is called the flutter determinant and must be solved for the flutter frequency "F and airspeed, UF •! As the determinant is complex, Re(D)=0 and Im(D)=0 Two equations with two unknowns Introduction to Aeroelasticity Solution •! The flutter determinant is nonlinear in " and U •! It can be solved using a Newton-Raphson scheme •! Given an initial value "i, Ui, a better value can be obtained from •! Where F=[Re(D) Im(D)]T •! The initial value of "i is usually one of the wind-off natural frequencies Introduction to Aeroelasticity Effect of Flexural Axis Since sinusoidal motion is assumed the Theodorsen equations are only valid at the flutter point (or when there is no damping at all) Therefore, the variation of the eigenvalues with airspeed cannot be calculated The flutter speeds calculated from Theodorsen are less conservative than the quasi-steady results Introduction to Aeroelasticity [...]... R2 za = x a + iy a = z + z y! ya! R! x! Introduction to Aeroelasticity -2R! 2R! xa! Singularities •! Theodorsen chose to use the following singularities: –! A free stream of speed U and zero angle of attack –! A pattern of sources of strength +2! on the top and surface of the flat plate, balanced by sources of strength -2! on the bottom surface –! A pattern of vortices +"! on the flat plate balanced... strength of the source distribution is defined by the wing’s motion Introduction to Aeroelasticity Wing motion •! Assume that the wing has pitch and plunge degrees of freedom •! The total upwash due to its motion is equal to ˙ ( ( )) w = ! U" + h + b( x1 + 1) ! x f "˙ •! where xf is the position of the flexural axis and x1 goes from -1 to +1 x x= b Introduction to Aeroelasticity and x is measured from the... the wing’s motion and the free stream, i.e !" = #w !n •! Where n is a unit vector normal to the surface and w is the external upwash Introduction to Aeroelasticity Impermeablity (2) •! Across the solid boundary of a closed object the source strength is given by #$ ! =" #n •! (assuming that the potential of the internal flow is constant) •! Therefore, !=w •! This means that the strength of the source... and wake are slits •! Different parts of the circle map to different parts of the wing y Circle upper surface -b b Circle lower surface Introduction to Aeroelasticity Outside circle Inside circle x Boundary conditions •! As with all attached flow aerodynamic problems there are two boundary conditions: –! Impermeability: the flow cannot cross the solid boundary –! Kutta condition: the flow must separate... using sources of strength 2! Introduction to Aeroelasticity About the wing and wake •! The wing is a flat plate with a source distribution that changes in time •! The +2! and -2! source contributions do not cancel each other out •! The wake of the wing is a flat line with vorticity that changes both in space and in time •! The +"! and -"! vorticity contributions do not cancel each other out Introduction. .. more complicated •! Here’s the result: mc = ! "Ub * ) 1 Introduction to Aeroelasticity #b % %$ 2 x0 + 1 x0 & ! ec (Vdx 0 2 x0 ! 1 x 0 ! 1 (' The nature of V •! V is a non-dimensional measure of vortex strength at a point x0 in ‘flat plate space’ •! As we’ve assumed that vortices don’t change strength as they travel downstream, V is a function of space •! In fact, it is stationary in value if we use... are using sources of strength 2!, the potential induced by a source at x1 , y1 and a sink at x1 , -y1 is given by 2 2 " % ( x $ x1 ) + ( y $ y1 ) ( d! ( x1,± y1 ) = ln ' * 2# '& ( x $ x1 ) 2 + ( y + y1 ) 2 )* •! The value of this potential does not change if we use non-dimensional coordinates 2 2 " % ( x $ x1 ) + ( y $ y1 ) ( d! ( x1,± y1 ) = ln ' * 2# '& ( x $ x1 ) 2 + ( y + y1 ) 2 *) Introduction to... x1, y1 ) = U# + h + b( x1 + 1) $ x f #˙ ln ' 2 2 *dx1 + 2" $1 '& ( x $ x1 ) + ( y + y1 ) *) 1 ( Introduction to Aeroelasticity ( )) After the integrations •! A long sequence of hardcore integration sessions has been censored Such scenes are unsuitable for 2nd year Master students and middle-aged engineering professors •! The result on the upper surface is: 2 b "˙ 2 ˙ ( x + 2) 1# x 2 (1) ! ( x, y ) = b(U"... plate balanced by identical but opposite -"! vortices in the wake Introduction to Aeroelasticity y 2! 2! Complete flowfield 2! x1,y1 Black dots: sources and sinks Red dots: vortices b +"! -"! b2/X0,0 X0,0 x1,-y1 -2! -2! Introduction to Aeroelasticity -2! The circle radius, b, is equal to the wing’s half-chord, b=c/2 x Complete flowfield – flat y plate Wing -b Wake b x Points inside the circle are transformed... + u = U cos ! + "x "x Introduction to Aeroelasticity Pressure difference •! The pressure on the upper surface is then 2 %1% #$ ' #$ ' pu = ! ") U + + * + Constant #x ( #t ( &2& •! And on the lower surface: 2 %1% #$ ' #$ ' pl = ! ") U ! ! * + Constant #x ( #t ( &2& •! The pressure difference is simply & $% $% ( & U $% $% ( = "2 # + + !p = pu " pl = "2 # U ' $x $ t ) ' b $x $t ) Introduction to Aeroelasticity