Unsteady Two-Dimensional Thin Airfoil Theory

Size: px
Start display at page:

Download "Unsteady Two-Dimensional Thin Airfoil Theory"

Transcription

1 Unsteady Two-Dimensional Thin Airfoil Theory General Formulation Consider a thin airfoil of infinite span and chord length c. The airfoil may have a small motion about its mean position. Let the x axis be aligned with the airfoil mean position and centered at its midchord O. The flow upstream of the airfoil may have small nonuniformities but its mean velocity U is uniform and in the x-direction. The velocity field can be written as V(x, y, t) = U + u (x, y, t) + u(x, y, t), () where u (x, y, t) represents the velocity of the imposed upstream nonuniformities, and u(x, y, t) stands for the perturbation velocity generated by the presence of the airfoil. The assumption of small airfoil motion and small upstream nonuniformities imply that u (x, y, t) /U << and u(x, y, t) /U <<. Normalize length with respect to c/2 Normalize velocity with respect to the freestream U Determine a velocity field u(x, y, t) such that Boundary Conditions: u = {u, v}. y =, < x < +: (a) v = specified (b) v + = v 2. y =, x > +: (a) pressure is continuous, p + = p (b) normal velocity is continuous, v + = v 3. u as (x 2 + y 2 ) 2 From the Euler Equations, Using condition 2, where u = u + u. Therefore, and ( u) T.E. = f( t), is the velocity jump at the trailing. u = u = (2) ( t + ) u ± = p ± x ρ x. (3) ( t + ) u =, (4) x u = f(x t), (5)

2 . Formulation in terms of sectionally analytic functions or u iv = 2πi As z x ±, < x < +, u iv = 2πi u i v dx (6) x z u x z dx + f(x t) 2πi x z dx (7) (u iv) ± = ± 2 u + 2πi c u x x dx + f(x t) 2πi x x dx (8) Taking the imaginary parts of both sides, v ± = 2π c u x x dx + f(x t) 2π x x dx (9) The real parts give u ± = ± 2 u, hence, u + = u = 2 u. If we specify the dependence on time f(x t) will be known. Therefore, we have the following integral equation for the unknown u: π c u x x dx = 2v f(x t) π x x dx () The solution to this singular integral equation is not unique. However, if we apply the Kutta condition, i.e., the solution must be finite at the trailing edge, we get u = π x + x c +x ( x 2v f(x ) t) dx x x π x x The double integral can be reduced to a single integral, and we have u = π x + x c + x x 2v + x x x dx x f(x t) x x The pressure jump, p = p + p, is given by integrating Equation 3. dx. () dx. (2) The lift L is given by p = ( u u T.E. ) x u dx (3) + + L = p dx = u dx 2( u) T.E. + Integrating the last integral by parts gives L = + u dx 2( u) T.E x u dx dx. (4) ( + x) u dx. (5)

3 Evaluation of the integrals is done by substitution of Equation 2 into Equation x u dx = 2 x v(x, t)dx + f(x t) x + x dx (6) ( + x) udx = x x x vdx + f(x x t) x + x (x + ) dx (7).2 Application of Kelvin s Theorem Kelvin s theorem states that the circulation around a circuit moving with the fluid is constant for a barotropic fluid with no viscosity and body forces. = u dx = + u dx + Substituting u from Equation 2 into (8) and using (6) 2 + f(x t)dx (8) + x x x v dx + f(x + t) x dx =. (9) This equation will determine the intensity of the vortex shedding in the wake..3 The Lift Formula Substituting (6 and 7) into (5), gives Noting that + + x L = 2 x v dx + + x +2 x v dx x + f(x t) x + x dx 2( u) T.E. f(x x t) x + x (x + ) dx (2) and using?? to show that x + x fdx (x + ) f(x t)dx = x x + fdx x = 3 f(x t)dx 2f( t), (2) f x 2 dx 2 + x x dx, (22)

4 gives the following expression for the lift, The first term, L = x x v dx 2 L q.s. = x 2 v dx + + x f x 2 dx. (23) x v dx (25) represents the quasi-steady lift. It is the lift obtained if the velocity, v, is time-independent. For example, if the airfoil is at angle of attack α to the upstream flow, then v = α and L q.s. = 2πα, which is the classical result for the lift coefficient of a thin airfoil. This is also the case when the time variation of v is small, then v and the vortex shedding in the wake is also negligible, f(x t). The second term, + L a.m. = 2 x 2 v dx 2,. (27) is proportional to the acceleration of the velocity v and therefore represents the apparent mass lift. This force dominates the response of the airfoil when v depends strongly on time. The last term, L f w = x 2 dx, (28) In dimensional form, we get, L q.s. = 2ρ U +c/2 c/2 c/2 + x c/2 x v dx = 2ρ U c x c x v dx. (25) Note that +c/2 c/2 c c/2 + x c/2 x dx = π c 2, x c x dx = π c 2. 2 In dimensional form, L a.m. = 2ρ ( c + c 2 )2 x 2 v dx = 2ρ x (c x ) v dx. (27) Note that + c x 2 dx = π 2, x (c x ) dx = π 8 c2. 4

5 represents the wake shedding contribution to the lift. Thus the lift can be written as the sum of three terms representing respectively, the quasi-steady lift, the apparent mass lift, and the wake induced lift, L = L q.s. + L a.m. + L w. (29) This result was first derived by von Karmen and Sears in The Moment Formula An elementary force df produces a moment with respect to the origin O, dm/o = xdf. The expression for the total moment is Substituting p from 3 gives M/O = + Integrating the last term by parts, M/O = + After considerable reduction, we obtain M/O = 2 + M/O = + x u dx + x u dx + 2 x pdx. (3) x + x udx dx. (3) + + x2 v dx x x 2 v dx 2 ( x 2 ) u dx. (32) f dx. (33) x2 As for the lift formula, the moment is the sum of three terms representing the quasi-steady moment, the apparent mass moment and the wake induced moment..5 Harmonic Time Dependence Equations (23 and 58)determine the lift and moment of an airfoil in terms of the velocity v and the distribution of the vortex shedding in the wake f(x t). The velocity v is specified but f(x t) must be determined from the time history of the motion. In many instances the airfoil is undergoing periodic oscillations or subject to periodic gust excitations. For these cases the velocity v is a periodic function of time and it is possible to consider the airfoil response to a single harmonic component v = ṽe iωt. In the general case it is also convenient to consider the Fourier transform v(x, t) = + ṽ(x, ω)e iωt dω, (34) and to calculate the response to every Fourier component ṽe iωt. This approach has the advantage of immediately determining the expression for the wake vortex distribution f(x t) = c o e i(ω(x t)), (35) 5

6 where c o is constant which can be readily determined from the Kelvin theorem condition by substituting (35) into (9), c o e iωt = 2 + +x v dx x x+ x eiωx dx. (36) The numerator in (36) is L q.s. and the integral in the denominator can be expressed in terms of Hankel functions, giving c o e iωt = L q.s. i π 2 H +(ω), (37) where H + (ω) = H () (ω) + ih () (ω). The wake-induced lift, L w, can also be evaluated by substituting (35) into (28), L H () (ω) w = L q.s. H + (ω). (38) Substituting (38) into (23) gives L = L q.s.c(ω) + L a.m., (39) where C(ω) = i H() (ω) H + (ω) is the complex conjugate of the Theodorsen function 3. ω, C(ω), and as ω, C(ω).5. Similarly, the expression for the moment becomes (4) It can be readily shown that as M/O = x2 v dx x x 2 v dx + 2 L q.s. ( C(ω)) (4) In what follows we apply the general formulas derived for the lift and moment to special cases. We first consider the cases of translatory and rotational oscillations as models for bending and torsional oscillations. We then study the case for a harmonic transverse gust impinging on the airfoil. Closed form analytical expressions for the lift and moments will be derived for these fundamental cases. The results will also be used to study the fundamental problems of a step change in the angle of attack and that of a sharp edged gust..5. Translatory Oscillations For a translatory oscillation the airfoil normal velocity v is independent of x and has the form v = ve iωt, where v, the magnitude of the oscillation, is constant. Using this expression for v in the various lift and moment formulas gives 3 Theodorsen considered a time dependence of the form exp(iωt). 6

7 Figure : Vector diagram of the real and imaginary parts of the Theodorsen function as the reduced frequency increases from to. L q.s. = πρcuv, (42) L a.m. = π 4 ρc2 v, (43) L w = L q.s. ( C(ω)), (44) L = L q.s.c(ω) π 4 ρc2 v (45) M q.s. /O = c 4 L q.s, (46) M a.m. /O =, (47) M w /O = c 4 L w, (48) M/O = c 4 L q.s.c(ω) (49) These results show that the center of pressure for both the the quasi-steady lift and the wake-induced lift is located at the quarter chord, and that the center of pressure for the apparent mass lift is located at the origin. 7

8 .5.2 Rotational Oscillations For an airfoil in rotational oscillation, the airfoil surface is y αx =, where α = αexp( iωt) is the airfoil angle with the x-axis, and α,the angular magnitude of the oscillation, is constant. The expression for the normal velocity is obtained by taking D /Dt(y α)x =, yielding v = αx + αu. It is seen that the velocity is the sum of two terms. The second term, αu, is independent of x and represents a translatory oscillation whose lift and moment are given by (42-49). Thus, we need only to calculate the expressions for the lift and moment in response to the first term αx by substituting this expression for v in the various lift and moment formulas. This gives Figure 2: Vector diagram of the real and imaginary parts of the lift coefficient of an airfoil in translatory oscillation. L q.s. = π 4 ρc2 U α, (5) L a.m. =, (5) L w = L q.s. ( C(ω)), (52) 8

9 L = L q.s.c(ω) (53) M q.s. /O =, (54) M a.m. /O = π 28 ρc4 α, (55) M w /O = c 4 L w, (56) M/O = c 4 L q.s. ( C(ω)) π 28 ρc4 α. (57) Figure 3: Vector diagram of the real and imaginary parts of the lift coefficient of an airfoil in rotational oscillation..5.3 Translatory and Rotational Oscillations For an airfoil in translatory and rotational oscillation hinged at the point O, the airfoil surface is y α(x x O ) =, v = α(x x O ) + αu. This is equivalent to the sum of a translatory oscillation, αu αx O, whose lift and moment are given by (42-49), and a rotaional oscillation about the origin whose lift and moment are given by (5-57). Note that the expression for the moment with respect to O is 9

10 .5.4 Step Change in Angle of Attack M/O = M/O x O L. (58) This problem examines the time evolution of the lift of an airfoil as the angle of attack α is suddenly increased from zero to α o, i.e., α = {, t <, α o, t >. This can be written in terms of the unit step function where The normal velocity is then given by y + u(t)α o x =, (59) u(t) = {, t <,, t >. v = u(t)α o U δ(t)α o x, (6) where δ(t) is the unit impulse or Dirac function and u(t) = δ(t). Define the unit step and Dirac functions as u(t) = πi c e iωt dω, (6) ω δ(t) = + e iωt dω. (62) 2π or, u(t) = 2 + π δ(t) = π + + sinωt dω, (63) ω cosωt dω. (64) As in previous results, we calculate the lift in two stages. We first consider the lift produced by the translatory part -u(t)α o U. The constant term in (63) gives a steady lift of πρcu 2 α o /2. The circulatory part of the lift produced by (α o Ue iωt )/(πω) is L C = ρcu 2 α o C(ω) (ω) e iωt (65). Hence, the lift produced by (α o Usinωt)/(πω) is the imaginary part of (65). circulatory lift can be obtained by integration, L C = ( ) [ πρcu 2 α o 2 Im([e iωt C(ω)] ω dω. ] The total (66)

11 The non-dimensional form of L C φ(t) = 2 Im [ e iωt) C(ω) ] ω dω. (67) is the Wagner function. By writing C(ω) in terms of its real and imaginary parts of C(ω) = Re [C(ω)] + iim [C(ω)], φ(t) = 2 π Im(C)cosωt ω For t <, the lift must be zero, therefore dω + π Re(C)sinωt dω. (68) ω 2 Im(C)cosωt dω = π ω π The wagner function can then be written as Re(C)sinω t dω. (69) ω φ(t) = 2 π Re(C)sinωt dω. (7) ω The limits for φ can be obtained as follows. We first note that for t >, 2 π sinωt dω =. (7) ω For t >, as t the integrand of (7) vanishes for small and finite values of ω; only large values of ω contribute to (7). Since for ω, C(ω).5 and using (7), we get lim t φ(t) =.5. For t, only vanishing values of ω contributes to (7). Since for ω, C(ω) and using (7), we get lim t φ(t) =. We give the following two approximations for the wagner function, φ(t) =.65e.455t.335e.3t φ(t) t + 2 t Airfoil in a Gust We consider a uniform flow U in the x direction with transverse sinusoidal gust in the y direction as shown in figure (5). Thus the unsteady velocity field is of the form u = a 2 je i(k x ωt) + u, (72) where k = (ωc)/(2u). Since we non-dimensionalize length and velocity with respect to c/2 and U, respectively, we get in non-dimensiosnal form k = ω. At the surface of the airfoil, the normal velocity must vanish, v = a 2 e ik x ωt) for y =, < x < +. (73)

12 Magnitude of Wagner Function Magnitude, φ(ω) Exact Approximation Approximation Time, s Figure 4: The Wagner function. Substituting (73) into (39) and noting that + + x x eik x = πj + (k ), + x2 e ikx = π J (k ), k where we have put J + = J + ij, we get L = πρcua 2 [J + (k )C(ω) i ω ] J (k ) e iωt. (74) k For a gust, k = ω. Using () for C(ω) and noting that J + H () J H + = 2i/(πω), we obtain the simple expression where L = πρcua 2 S(ω)e iωt, (75) S(ω) = 2 πωh + (ω). (76) The function S is the complex conjugate of the well known Sears function 4. Traditionally, the Sears function is plotted as a vector diagram of the real and imaginary parts versus the reduced frequency as shown in figure(6). The Sears function is widely used in applications. A good approximation for its magnitude is given by S = + 2πω. (77) 4 Sears considered a time dependence of the form exp(iωt). 2

13 Figure 5: Thin airfoil in a transverse gust..5.6 Sharp-Edged Gust An incident sharp-edged gust is defined as a traveling vertical gust where v g = {, x < Ut c/2 v o, x > Ut c/2. (78) As for the case of a step change in angle of attack, we use the unit step function to represent the gust velocity. Equation (78) can be written as v g = v o u(x Ut + c 2 ) (79) Using the expression (63) for the unit step function and following the procedure developed in.5.4, gives the expression for the lift [ L = πρcuv o πi c Like the previous case, we can rewrite the lift as L = πρcuv o ψ(t), S(ω)e iω(t) ] ω dω, (8) where ψ(t) = πi c S(ω)e iω(t c 2U ) dω (8) ω is the Kussner function. It is convenient to cast the lift in terms of a non-singular integrand ψ(t) = 2 π + Re [ S(ω)e iω] sinωt ω dω (82) For t >, as t the integrand of (7) vanishes for small and finite values of ω; For large values of ω the Sears function vanishes. Hence, lim t ψ(t) =. For t, only 3

14 Figure 6: Vector diagram of the real and imaginary parts of the Sears function as the reduced frequency increases from to. vanishing values of ω contributes to (7). Since for ω, S(ω) and using (7), we get lim t ψ(t) =. The following approximations for ψ were derived ψ(t) =.5e.3t.5e t t 2 + t t t +.8 (83) Figure 8 shows the variation of the Kussner function and a comparison with the two approximations. Remember, the normalization for time is t = 2 Unsteady Subsonic Flows t. c/(2u) The theory developed for incompressible two dimensional unsteady flow paved the way to our understanding of unsteady flow phenomena and its application to aeroelasticity and aeroacoustics. As flight speed increased, it became necessary to account for the effects of Mach number or more complex incident disturbances. A good starting point is to begin with the linearized Euler equations about a mean uniform flow, namely, and D o Dt ρ + ρ o u =, (84) D o ρ o u = p. (85) Dt 4

15 Magnitude of Sears Function.9 Magnitude, S(ω) Exact Approximation Frequency, ω Figure 7: Magnitude of the Sears function. We further assume the flow to be isentropic and hence p = c 2 oρ, where c is the speed of sound. Taking the material derivative of (84 and the divergence of(85 and eliminating the velocity, we get the convected wave equation ( c 2 o D 2 o Dt 2 2 Expanding (86) and arranging the terms gives ( M 2 ) 2 p + 2 p + 2 p 2 M x 2 x 2 2 x 2 3 c o where M is the Mach number. We now introduce the Prandtl-Glauert coordinates: ) p =. (86) 2 p x t c 2 o 2 p =. (87) t2 x = x, x 2 = x 2 β, x 3 = x 3 β, (88) where β = M 2. Equation (87) reduces to 2 p 2 M p c o β 2 x t 2 p =, (89) c 2 oβ 2 t2 where is the del operator in the Prandtl-Glauert coordinates. We now consider the case of time-harmonic dependence of the pressure of the form p = p(x)e iωt. (9) 5

16 Magnitude of Kussner Function Magnitude, ψ(ω) Exact Cauchy Approximation Approximation 2 Exact Sin/Cos Time, s Figure 8: The Kussner function. Substituting (9) into (86) gives or, ( M 2 ) 2 p [ ( ω + im ] ) p =. (9) c 2 o c x x p + 2 p + 2iω M p x 2 2 x 2 3 c o ] [ 2 + 2i ωm β 2 c o x + ω2 β 2 c 2 o In order to simplify (93), we introduce the new variable Note that 2 p x 2 The governing equation (93) can then be written Taking α = Mω c o β 2 2 p + 2iα p α 2 p + 2iMω x c o β 2 gives x + ω2 c 2 o 2 p =. (92) t2 p =, (93) p = p(x)e iαx.. (94) ( ) p p = + iα p e iα x x x (95) ( 2 p = + 2iα p ) α 2 p e iα x x 2 x (96) p + 2ωMα x c o β 2 p + ω2 p =. (97) c 2 2 oβ 6

17 or 2 p + ( ω 2 c 2 β ω2 M 2 2 c 2 oβ + 2ω2 M 2 ) 4 c 2 oβ 4 p =, (98) ( 2 + K 2 ω ) p =, (99) where we have put K ω = ω/(c o β 2 ). This important result shows that the linearized Euler equations reduce the Helmholtz equation as in acoustics. Note that the velocity potentioal defined by u = φ () also satisfies the convected wave equation (86). Following the same procedure which led to (99) for the pressure and writing φ = φexp [ i (K ω M x + ωt)], we get ( ) 2 + Kω 2 φ =. () 2. Fundamental Solution The Green s function is solution to [ ( ω + im ] ) p = 4πδ(x x c 2 ), (2) o c x where x is the observation point and x is the source point. Expanding (2), dividing both sides by β 2 and noting that δ(x x ) = δ( x x )/β 2, we get The solution to (3) is ( 2 + K 2 ω ) p = 4πδ( x x )e ik ωm x, (3) eikω( r+m x ) p =, (4) r where r = x x. The expression for p can be readily obtained as p = ei{k ω[ r M( x x )] ωt}. (5) r Or in terms of the physical variables, { ] } K ω r[ ( M 2 sin 2 θ) 2 M cos θ ωt p = ei r ( M 2 sin 2 θ ), (6) 2 where r = x x and θ = sin x x 2 2/r is the angle between the observation point direction and the x -axis. For M =, (6) reduces to the expression for the pressure from a source without mean motion. The effect of the Mach number is to distort the pressure magnitude and phase which become dependent on the angle θ. 7

18 2.2 Subsonic Two-Dimensional Unsteady Airfoil Theory Consider a thin two-dimensional airfoil of infinite span and chord length c. The airfoil may have a small motion about its mean position. Let the x axis be aligned with the airfoil mean position and centered at its midchord O. The flow upstream of the airfoil may have small nonuniformities but its mean velocity U is uniform and in the x -direction. The x 2 axis is in the direction of the airfoil span and the x 3 axis is perpendicular to the airfoil in the upward direction. In what follows we non-dimensionalize length with respect to c/2. The velocity field can be written as V(x, y, t) = U + u (x, t) + u(x, t), (7) where u (x, t) represents the velocity of the imposed upstream nonuniformities, and u(x, t) stands for the perturbation velocity generated by the presence of the airfoil. The assumption of small airfoil motion and small upstream nonuniformities imply that u (x, t) /U << and u(x, t) /U <<. We introduce the potential function u(x, t) = φ. (8) Then the unsteady pressure We introduce the reduced frequency and the nondimensional form of K ω D p = ρ φ. (9) Dt ω = ωc 2U () The boundary conditions are as follows The pressure p vanishes at infinity. K = ω M β 2 () The vertical velocity φ/ x 3 is specified at c/2 < x < c/2, x 2 =. p is continuous in the wake. We consider time-harmonic dependence. In addition, for two-dimensional airfoils of infinite span it is always possible to consider a Fourier expansion in the span direction and to consider a single component of the form exp(ik 2 x 2 ). This allows us to factor out the time and the x 2 dependence by writing φ = ( c 2 u )ϕ( x, x 3 )exp [i (k 2 x 2 K M x ωt)], (2) where u is a representative value for the upwash u 3 = u u 3 exp[i(k 2 x 2 K M x ωt)]. The equation for ϕ can be readily obtained from (), giving 8

19 ( 2 + K 2) ϕ =. (3) Here, 2 2 = 2 x 2 is the two-dimensional Laplacian operator in the Prandtl-Glauert plane, i.e., and we have put K 2 = Kω 2 K2 2 with K 2 = k 2 /β. + 2 x 2 3 This reduces the boundary-value problem for the unsteady infinite span airfoil in subsonic flow to that of a two-dimensional boundary-value problem where the governing equation is the two-dimensional Helmholtz equation. The expression for the pressure is given by The boundary conditions are p(x, t) = (ρ Uu ) p( x, x 3 )e i(k 2x 2 ωt), (4) p( x, x 3 ) = (i ω β ϕ ϕ )e ik M x. 2 x (5) ϕ x 3 = β eik ωm x u 3 (6) i ω β ϕ ϕ 2 x = for x >, (7) ϕ = as x. (8) Equations (3, 6 8) state that aerodynamic problem of an infinite plate vibrating or subject to a three-dimensional gust can be formulated in terms of a two-dimensional problem. The plate vibratory motion must be expanded in terms of its Fourier components in the x 2 direction, i.e., exp(ik 2 x 2 ). 2.3 Integral Equation Formulation The elementary solution ϕ(α) = e iα x α 2 K 2 x 3, satisfyies the Helmholtz equation (3 ) for any α. Hence, the general solution is given by + ϕ = ± g(α)e iα x α 2 K 2 x 3 dα. (9) Note that g(α) is unknown. Taking the derivative with respect to x 3 gives ϕ x 3 = Now along the body, x 3 =, the equation for ϕ and p are + α2 K 2 g(α) e iα x α 2 K 2 x 3 dα. (2) 9

20 + ϕ = 2 g(α)e iα x dα. (2) p = 2ie ik M x + f(α)e iα x dα, (22) where we have set f(α) = ( ω α)g(α) (23) β2 This looks like a Fourier transform. Therefore, can invert it to solve for f(α): f(α) = i + pe i(α K M) x d x. (24) 4π Note that the limits of integration collapsed to this region because outside of (, +), p. Using ( 2, 22, 23), we get Or, where u 3( x ) = iβ + 4π e ik M x α2 K c 2 u 3( x ) = The solution is a series of the form where cos θ = x. eiα x ω α β p e i(α K ωm) x d x dα (25) p K( x x ) d x, (26) K(ξ) = iβ + 4π c α2 K 2 ω α ei(α KM)ξ dα. (27) β 2 p = 2 A o cot ( ) θ + 2 It is remarkable to note that equation (25) can be cast in the form A m sin(mθ), (28) m= where U 3 ( x ) = i 4π c + α2 K 2 eiα x ω α β 2 + P e iα x d x dα, (29) 2

21 U 3 ( x ) = β u 3( x )e ik M x (3) P = pe ik M x. (3) Therefore for the same parameters K and ω /β 2 P is only function of the modified upwash U 3. 2

22 Figure 9: Vector diagram showing the real and imaginary parts of the response function S(k, k 3, M = ) versus k for a transverse gust at various M. Lines of constant k are shown for k =.2,.6,.,.5, 2., 3., 4.,

23 Figure : Vector diagram showing the real and imaginary parts of the response function S(k,, M) versus k for an oblique gust at various k 3. Figure : Vector diagram showing the real and imaginary parts of the response function S(k, k 3, M =.8) versus k for an oblique gust at various k 3. 23

Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics

Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/

More information

Research Article Numerical Study of Flutter of a Two-Dimensional Aeroelastic System

Research Article Numerical Study of Flutter of a Two-Dimensional Aeroelastic System ISRN Mechanical Volume 213, Article ID 127123, 4 pages http://dx.doi.org/1.1155/213/127123 Research Article Numerical Study of Flutter of a Two-Dimensional Aeroelastic System Riccy Kurniawan Department

More information

Aeroacoustic and Aerodynamics of Swirling Flows*

Aeroacoustic and Aerodynamics of Swirling Flows* Aeroacoustic and Aerodynamics of Swirling Flows* Hafiz M. Atassi University of Notre Dame * supported by ONR grant and OAIAC OVERVIEW OF PRESENTATION Disturbances in Swirling Flows Normal Mode Analysis

More information

Bernoulli's equation: 1 p h t p t. near the far from plate the plate. p u

Bernoulli's equation: 1 p h t p t. near the far from plate the plate. p u UNSTEADY FLOW let s re-visit the Kutta condition when the flow is unsteady: p u p l Bernoulli's equation: 2 φ v 1 + + = () = + p h t p t 2 2 near the far from plate the plate as a statement of the Kutta

More information

Fundamentals of Fluid Dynamics: Waves in Fluids

Fundamentals of Fluid Dynamics: Waves in Fluids Fundamentals of Fluid Dynamics: Waves in Fluids Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/ tzielins/ Institute

More information

Lifting Airfoils in Incompressible Irrotational Flow. AA210b Lecture 3 January 13, AA210b - Fundamentals of Compressible Flow II 1

Lifting Airfoils in Incompressible Irrotational Flow. AA210b Lecture 3 January 13, AA210b - Fundamentals of Compressible Flow II 1 Lifting Airfoils in Incompressible Irrotational Flow AA21b Lecture 3 January 13, 28 AA21b - Fundamentals of Compressible Flow II 1 Governing Equations For an incompressible fluid, the continuity equation

More information

Fourier series. Complex Fourier series. Positive and negative frequencies. Fourier series demonstration. Notes. Notes. Notes.

Fourier series. Complex Fourier series. Positive and negative frequencies. Fourier series demonstration. Notes. Notes. Notes. Fourier series Fourier series of a periodic function f (t) with period T and corresponding angular frequency ω /T : f (t) a 0 + (a n cos(nωt) + b n sin(nωt)), n1 Fourier series is a linear sum of cosine

More information

Review of Fundamental Equations Supplementary notes on Section 1.2 and 1.3

Review of Fundamental Equations Supplementary notes on Section 1.2 and 1.3 Review of Fundamental Equations Supplementary notes on Section. and.3 Introduction of the velocity potential: irrotational motion: ω = u = identity in the vector analysis: ϕ u = ϕ Basic conservation principles:

More information

High Speed Aerodynamics. Copyright 2009 Narayanan Komerath

High Speed Aerodynamics. Copyright 2009 Narayanan Komerath Welcome to High Speed Aerodynamics 1 Lift, drag and pitching moment? Linearized Potential Flow Transformations Compressible Boundary Layer WHAT IS HIGH SPEED AERODYNAMICS? Airfoil section? Thin airfoil

More information

2004 ASME Rayleigh Lecture

2004 ASME Rayleigh Lecture 24 ASME Rayleigh Lecture Fluid-Structure Interaction and Acoustics Hafiz M. Atassi University of Notre Dame Rarely does one find a mass of analysis without illustrations from experience. Rayleigh Analysis

More information

Research Article Nonlinear Characteristics of Helicopter Rotor Blade Airfoils: An Analytical Evaluation

Research Article Nonlinear Characteristics of Helicopter Rotor Blade Airfoils: An Analytical Evaluation Mathematical Problems in Engineering olume 2, Article ID 5858, 9 pages http://dx.doi.org/.55/2/5858 Research Article Nonlinear Characteristics of Helicopter Rotor Blade Airfoils: An Analytical Evaluation

More information

Equations of linear stellar oscillations

Equations of linear stellar oscillations Chapter 4 Equations of linear stellar oscillations In the present chapter the equations governing small oscillations around a spherical equilibrium state are derived. The general equations were presented

More information

Aeroelastic Analysis Of Membrane Wings

Aeroelastic Analysis Of Membrane Wings Aeroelastic Analysis Of Membrane Wings Soumitra P. Banerjee and Mayuresh J. Patil Virginia Polytechnic Institute and State University, Blacksburg, Virginia 46-3 The physics of flapping is very important

More information

Aeroelasticity. Lecture 9: Supersonic Aeroelasticity. G. Dimitriadis. AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 9

Aeroelasticity. Lecture 9: Supersonic Aeroelasticity. G. Dimitriadis. AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 9 Aeroelasticity Lecture 9: Supersonic Aeroelasticity G. Dimitriadis AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 9 1 Introduction All the material presented up to now concerned incompressible

More information

Limit Cycle Oscillations of a Typical Airfoil in Transonic Flow

Limit Cycle Oscillations of a Typical Airfoil in Transonic Flow Limit Cycle Oscillations of a Typical Airfoil in Transonic Flow Denis B. Kholodar, United States Air Force Academy, Colorado Springs, CO 88 Earl H. Dowell, Jeffrey P. Thomas, and Kenneth C. Hall Duke University,

More information

Steady waves in compressible flow

Steady waves in compressible flow Chapter Steady waves in compressible flow. Oblique shock waves Figure. shows an oblique shock wave produced when a supersonic flow is deflected by an angle. Figure.: Flow geometry near a plane oblique

More information

Continuum Mechanics Lecture 7 Theory of 2D potential flows

Continuum Mechanics Lecture 7 Theory of 2D potential flows Continuum Mechanics ecture 7 Theory of 2D potential flows Prof. http://www.itp.uzh.ch/~teyssier Outline - velocity potential and stream function - complex potential - elementary solutions - flow past a

More information

Chapter 2: Complex numbers

Chapter 2: Complex numbers Chapter 2: Complex numbers Complex numbers are commonplace in physics and engineering. In particular, complex numbers enable us to simplify equations and/or more easily find solutions to equations. We

More information

FLOW OVER A GIVEN PROFILE IN A CHANNEL WITH DYNAMICAL EFFECTS

FLOW OVER A GIVEN PROFILE IN A CHANNEL WITH DYNAMICAL EFFECTS Proceedings of Czech Japanese Seminar in Applied Mathematics 2004 August 4-7, 2004, Czech Technical University in Prague http://geraldine.fjfi.cvut.cz pp. 63 72 FLOW OVER A GIVEN PROFILE IN A CHANNEL WITH

More information

Incompressible Flow Over Airfoils

Incompressible Flow Over Airfoils Chapter 7 Incompressible Flow Over Airfoils Aerodynamics of wings: -D sectional characteristics of the airfoil; Finite wing characteristics (How to relate -D characteristics to 3-D characteristics) How

More information

Theory of Ship Waves (Wave-Body Interaction Theory) Quiz No. 2, April 25, 2018

Theory of Ship Waves (Wave-Body Interaction Theory) Quiz No. 2, April 25, 2018 Quiz No. 2, April 25, 2018 (1) viscous effects (2) shear stress (3) normal pressure (4) pursue (5) bear in mind (6) be denoted by (7) variation (8) adjacent surfaces (9) be composed of (10) integrand (11)

More information

Wings and Bodies in Compressible Flows

Wings and Bodies in Compressible Flows Wings and Bodies in Compressible Flows Prandtl-Glauert-Goethert Transformation Potential equation: 1 If we choose and Laplace eqn. The transformation has stretched the x co-ordinate by 2 Values of at corresponding

More information

Helicopter Rotor Unsteady Aerodynamics

Helicopter Rotor Unsteady Aerodynamics Helicopter Rotor Unsteady Aerodynamics Charles O Neill April 26, 2002 Helicopters can t fly; they re just so ugly the earth repels them. 1 Introduction Unsteady helicopter rotor aerodynamics are complicated.

More information

Aeroelasticity. Lecture 7: Practical Aircraft Aeroelasticity. G. Dimitriadis. AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 7

Aeroelasticity. Lecture 7: Practical Aircraft Aeroelasticity. G. Dimitriadis. AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 7 Aeroelasticity Lecture 7: Practical Aircraft Aeroelasticity G. Dimitriadis AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 7 1 Non-sinusoidal motion Theodorsen analysis requires that

More information

Oscillations and Waves

Oscillations and Waves Oscillations and Waves Somnath Bharadwaj and S. Pratik Khastgir Department of Physics and Meteorology IIT Kharagpur Module : Oscillations Lecture : Oscillations Oscillations are ubiquitous. It would be

More information

18.325: Vortex Dynamics

18.325: Vortex Dynamics 8.35: Vortex Dynamics Problem Sheet. Fluid is barotropic which means p = p(. The Euler equation, in presence of a conservative body force, is Du Dt = p χ. This can be written, on use of a vector identity,

More information

INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad

INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad - 500 043 AERONAUTICAL ENGINEERING TUTORIAL QUESTION BANK Course Name : LOW SPEED AERODYNAMICS Course Code : AAE004 Regulation : IARE

More information

AE301 Aerodynamics I UNIT B: Theory of Aerodynamics

AE301 Aerodynamics I UNIT B: Theory of Aerodynamics AE301 Aerodynamics I UNIT B: Theory of Aerodynamics ROAD MAP... B-1: Mathematics for Aerodynamics B-: Flow Field Representations B-3: Potential Flow Analysis B-4: Applications of Potential Flow Analysis

More information

Given the water behaves as shown above, which direction will the cylinder rotate?

Given the water behaves as shown above, which direction will the cylinder rotate? water stream fixed but free to rotate Given the water behaves as shown above, which direction will the cylinder rotate? ) Clockwise 2) Counter-clockwise 3) Not enough information F y U 0 U F x V=0 V=0

More information

1. Fluid Dynamics Around Airfoils

1. Fluid Dynamics Around Airfoils 1. Fluid Dynamics Around Airfoils Two-dimensional flow around a streamlined shape Foces on an airfoil Distribution of pressue coefficient over an airfoil The variation of the lift coefficient with the

More information

Fourier transforms, Generalised functions and Greens functions

Fourier transforms, Generalised functions and Greens functions Fourier transforms, Generalised functions and Greens functions T. Johnson 2015-01-23 Electromagnetic Processes In Dispersive Media, Lecture 2 - T. Johnson 1 Motivation A big part of this course concerns

More information

Aerodynamic Rotor Model for Unsteady Flow and Wake Impact

Aerodynamic Rotor Model for Unsteady Flow and Wake Impact Aerodynamic Rotor Model for Unsteady Flow and Wake Impact N. Bampalas, J. M. R. Graham Department of Aeronautics, Imperial College London, Prince Consort Road, London, SW7 2AZ June 28 1 (Steady Kutta condition)

More information

MATH 566: FINAL PROJECT

MATH 566: FINAL PROJECT MATH 566: FINAL PROJECT December, 010 JAN E.M. FEYS Complex analysis is a standard part of any math curriculum. Less known is the intense connection between the pure complex analysis and fluid dynamics.

More information

Damped Harmonic Oscillator

Damped Harmonic Oscillator Damped Harmonic Oscillator Wednesday, 23 October 213 A simple harmonic oscillator subject to linear damping may oscillate with exponential decay, or it may decay biexponentially without oscillating, or

More information

Copyright 2007 N. Komerath. Other rights may be specified with individual items. All rights reserved.

Copyright 2007 N. Komerath. Other rights may be specified with individual items. All rights reserved. Low Speed Aerodynamics Notes 5: Potential ti Flow Method Objective: Get a method to describe flow velocity fields and relate them to surface shapes consistently. Strategy: Describe the flow field as the

More information

Math 575-Lecture Failure of ideal fluid; Vanishing viscosity. 1.1 Drawbacks of ideal fluids. 1.2 vanishing viscosity

Math 575-Lecture Failure of ideal fluid; Vanishing viscosity. 1.1 Drawbacks of ideal fluids. 1.2 vanishing viscosity Math 575-Lecture 12 In this lecture, we investigate why the ideal fluid is not suitable sometimes; try to explain why the negative circulation appears in the airfoil and introduce the vortical wake to

More information

State-Space Representation of Unsteady Aerodynamic Models

State-Space Representation of Unsteady Aerodynamic Models State-Space Representation of Unsteady Aerodynamic Models -./,1231-44567 &! %! $! #! "! x α α k+1 A ERA x = 1 t α 1 α x C L (k t) = C ERA C Lα C L α α α ERA Model fast dynamics k + B ERA t quasi-steady

More information

So far we have derived two electrostatic equations E = 0 (6.2) B = 0 (6.3) which are to be modified due to Faraday s observation,

So far we have derived two electrostatic equations E = 0 (6.2) B = 0 (6.3) which are to be modified due to Faraday s observation, Chapter 6 Maxwell Equations 6.1 Maxwell equations So far we have derived two electrostatic equations and two magnetostatics equations E = ρ ɛ 0 (6.1) E = 0 (6.2) B = 0 (6.3) B = µ 0 J (6.4) which are to

More information

AERODYNAMIC ANALYSIS OF THE HELICOPTER ROTOR USING THE TIME-DOMAIN PANEL METHOD

AERODYNAMIC ANALYSIS OF THE HELICOPTER ROTOR USING THE TIME-DOMAIN PANEL METHOD 7 TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES AERODYNAMIC ANALYSIS OF THE HELICOPTER ROTOR USING THE TIME-DOMAIN PANEL METHOD Seawook Lee*, Hyunmin Choi*, Leesang Cho*, Jinsoo Cho** * Department

More information

Linear second-order differential equations with constant coefficients and nonzero right-hand side

Linear second-order differential equations with constant coefficients and nonzero right-hand side Linear second-order differential equations with constant coefficients and nonzero right-hand side We return to the damped, driven simple harmonic oscillator d 2 y dy + 2b dt2 dt + ω2 0y = F sin ωt We note

More information

THE FOURIER TRANSFORM (Fourier series for a function whose period is very, very long) Reading: Main 11.3

THE FOURIER TRANSFORM (Fourier series for a function whose period is very, very long) Reading: Main 11.3 THE FOURIER TRANSFORM (Fourier series for a function whose period is very, very long) Reading: Main 11.3 Any periodic function f(t) can be written as a Fourier Series a 0 2 + a n cos( nωt) + b n sin n

More information

To study the motion of a perfect gas, the conservation equations of continuity

To study the motion of a perfect gas, the conservation equations of continuity Chapter 1 Ideal Gas Flow The Navier-Stokes equations To study the motion of a perfect gas, the conservation equations of continuity ρ + (ρ v = 0, (1.1 momentum ρ D v Dt = p+ τ +ρ f m, (1.2 and energy ρ

More information

Thin airfoil theory. Chapter Compressible potential flow The full potential equation

Thin airfoil theory. Chapter Compressible potential flow The full potential equation hapter 4 Thin airfoil theory 4. ompressible potential flow 4.. The full potential equation In compressible flow, both the lift and drag of a thin airfoil can be determined to a reasonable level of accuracy

More information

c 2011 Craig G. Merrett

c 2011 Craig G. Merrett c 011 Craig G. Merrett AERO-SERVO-VISCOELASTICITY THEORY: LIFTING SURFACES, PLATES, VELOCITY TRANSIENTS, FLUTTER, AND INSTABILITY BY CRAIG G. MERRETT DISSERTATION Submitted in partial fulfillment of the

More information

Partial Differential Equations for Engineering Math 312, Fall 2012

Partial Differential Equations for Engineering Math 312, Fall 2012 Partial Differential Equations for Engineering Math 312, Fall 2012 Jens Lorenz July 17, 2012 Contents Department of Mathematics and Statistics, UNM, Albuquerque, NM 87131 1 Second Order ODEs with Constant

More information

Transient Phenomena in Quantum Bound States Subjected to a Sudden Perturbation

Transient Phenomena in Quantum Bound States Subjected to a Sudden Perturbation Symmetry, Integrability and Geometry: Methods and Applications Vol. (5), Paper 3, 9 pages Transient Phenomena in Quantum Bound States Subjected to a Sudden Perturbation Marcos MOSHINSKY and Emerson SADURNÍ

More information

[N175] Development of Combined CAA-CFD Algorithm for the Efficient Simulation of Aerodynamic Noise Generation and Propagation

[N175] Development of Combined CAA-CFD Algorithm for the Efficient Simulation of Aerodynamic Noise Generation and Propagation The 32nd International Congress and Exposition on Noise Control Engineering Jeju International Convention Center, Seogwipo, Korea, August 25-28, 2003 [N175] Development of Combined CAA-CFD Algorithm for

More information

CHAPTER 5. RUDIMENTS OF HYDRODYNAMIC INSTABILITY

CHAPTER 5. RUDIMENTS OF HYDRODYNAMIC INSTABILITY 1 Lecture Notes on Fluid Dynamics (1.63J/.1J) by Chiang C. Mei, 00 CHAPTER 5. RUDIMENTS OF HYDRODYNAMIC INSTABILITY References: Drazin: Introduction to Hydrodynamic Stability Chandrasekar: Hydrodynamic

More information

Wave Phenomena Physics 15c

Wave Phenomena Physics 15c Wave Phenomena Physics 5c Lecture Fourier Analysis (H&L Sections 3. 4) (Georgi Chapter ) What We Did Last ime Studied reflection of mechanical waves Similar to reflection of electromagnetic waves Mechanical

More information

Wave Phenomena Physics 15c. Lecture 11 Dispersion

Wave Phenomena Physics 15c. Lecture 11 Dispersion Wave Phenomena Physics 15c Lecture 11 Dispersion What We Did Last Time Defined Fourier transform f (t) = F(ω)e iωt dω F(ω) = 1 2π f(t) and F(w) represent a function in time and frequency domains Analyzed

More information

UNSTEADY LIFT OF THICK AIRFOILS IN INCOMPRESSIBLE TURBULENT FLOW

UNSTEADY LIFT OF THICK AIRFOILS IN INCOMPRESSIBLE TURBULENT FLOW The Pennsylvania State University The Graduate School Graduate Program in Acoustics UNSTEADY LIFT OF THICK AIRFOILS IN INCOMPRESSIBLE TURBULENT FLOW A Dissertation in Acoustics by Peter D. Lysak c 2011

More information

Analysis of Structural Mistuning Effects on Bladed Disc Vibrations Including Aerodynamic Damping

Analysis of Structural Mistuning Effects on Bladed Disc Vibrations Including Aerodynamic Damping Purdue University Purdue e-pubs International Compressor Engineering Conference School of Mechanical Engineering 24 Analysis of Structural Mistuning Effects on Bladed Disc Vibrations Including Aerodynamic

More information

4. Complex Oscillations

4. Complex Oscillations 4. Complex Oscillations The most common use of complex numbers in physics is for analyzing oscillations and waves. We will illustrate this with a simple but crucially important model, the damped harmonic

More information

Unsteady Flow Over an Aerofoil. Thomas Roger Brickell. Supervisor Dr Jean - Luc Thiffeault.

Unsteady Flow Over an Aerofoil. Thomas Roger Brickell. Supervisor Dr Jean - Luc Thiffeault. nsteady Flow Over an Aerofoil Thomas Roger Brickell. Supervisor Dr Jean - Luc Thiffeault. September 3, 4 Acknowledgements I would like to thank my supervisor Dr Jean - Luc Thiffeault for his invaluable

More information

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t) IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common

More information

How many initial conditions are required to fully determine the general solution to a 2nd order linear differential equation?

How many initial conditions are required to fully determine the general solution to a 2nd order linear differential equation? How many initial conditions are required to fully determine the general solution to a 2nd order linear differential equation? (A) 0 (B) 1 (C) 2 (D) more than 2 (E) it depends or don t know How many of

More information

2 The incompressible Kelvin-Helmholtz instability

2 The incompressible Kelvin-Helmholtz instability Hydrodynamic Instabilities References Chandrasekhar: Hydrodynamic and Hydromagnetic Instabilities Landau & Lifshitz: Fluid Mechanics Shu: Gas Dynamics 1 Introduction Instabilities are an important aspect

More information

An Internet Book on Fluid Dynamics. Joukowski Airfoils

An Internet Book on Fluid Dynamics. Joukowski Airfoils An Internet Book on Fluid Dynamics Joukowski Airfoils One of the more important potential flow results obtained using conformal mapping are the solutions of the potential flows past a family of airfoil

More information

Webster s horn model on Bernoulli flow

Webster s horn model on Bernoulli flow Webster s horn model on Bernoulli flow Aalto University, Dept. Mathematics and Systems Analysis January 5th, 2018 Incompressible, steady Bernoulli principle Consider a straight tube Ω R 3 havin circular

More information

Vibrations and waves: revision. Martin Dove Queen Mary University of London

Vibrations and waves: revision. Martin Dove Queen Mary University of London Vibrations and waves: revision Martin Dove Queen Mary University of London Form of the examination Part A = 50%, 10 short questions, no options Part B = 50%, Answer questions from a choice of 4 Total exam

More information

In the present work the diffraction of plane electromagnetic waves by an impedance loaded parallel plate waveguide formed by a twopart

In the present work the diffraction of plane electromagnetic waves by an impedance loaded parallel plate waveguide formed by a twopart Progress In Electromagnetics Research, PIER 6, 93 31, 6 A HYBRID METHOD FOR THE SOLUTION OF PLANE WAVE DIFFRACTION BY AN IMPEDANCE LOADED PARALLEL PLATE WAVEGUIDE G. Çınar Gebze Institute of Technology

More information

Continuity Equation for Compressible Flow

Continuity Equation for Compressible Flow Continuity Equation for Compressible Flow Velocity potential irrotational steady compressible Momentum (Euler) Equation for Compressible Flow Euler's equation isentropic velocity potential equation for

More information

Time-domain modelling of the wind forces acting on a bridge deck with indicial functions

Time-domain modelling of the wind forces acting on a bridge deck with indicial functions POLITECNICO DI MILANO Facoltà di Ingegneria Civile, Ambientale e Territoriale Corso di Laurea Specialistica in Ingegneria Civile Indirizzo Strutture Time-domain modelling of the wind forces acting on a

More information

Flight Vehicle Terminology

Flight Vehicle Terminology Flight Vehicle Terminology 1.0 Axes Systems There are 3 axes systems which can be used in Aeronautics, Aerodynamics & Flight Mechanics: Ground Axes G(x 0, y 0, z 0 ) Body Axes G(x, y, z) Aerodynamic Axes

More information

221B Lecture Notes on Resonances in Classical Mechanics

221B Lecture Notes on Resonances in Classical Mechanics 1B Lecture Notes on Resonances in Classical Mechanics 1 Harmonic Oscillators Harmonic oscillators appear in many different contexts in classical mechanics. Examples include: spring, pendulum (with a small

More information

SPC Aerodynamics Course Assignment Due Date Monday 28 May 2018 at 11:30

SPC Aerodynamics Course Assignment Due Date Monday 28 May 2018 at 11:30 SPC 307 - Aerodynamics Course Assignment Due Date Monday 28 May 2018 at 11:30 1. The maximum velocity at which an aircraft can cruise occurs when the thrust available with the engines operating with the

More information

UNIVERSITY of LIMERICK

UNIVERSITY of LIMERICK UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4607 SEMESTER: Autumn 2012-13 MODULE TITLE: Introduction to Fluids DURATION OF

More information

Hankel Transforms - Lecture 10

Hankel Transforms - Lecture 10 1 Introduction Hankel Transforms - Lecture 1 The Fourier transform was used in Cartesian coordinates. If we have problems with cylindrical geometry we will need to use cylindtical coordinates. Thus suppose

More information

( ) f (k) = FT (R(x)) = R(k)

( ) f (k) = FT (R(x)) = R(k) Solving ODEs using Fourier Transforms The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d 2 dx + q f (x) = R(x)

More information

kg meter ii) Note the dimensions of ρ τ are kg 2 velocity 2 meter = 1 sec 2 We will interpret this velocity in upcoming slides.

kg meter ii) Note the dimensions of ρ τ are kg 2 velocity 2 meter = 1 sec 2 We will interpret this velocity in upcoming slides. II. Generalizing the 1-dimensional wave equation First generalize the notation. i) "q" has meant transverse deflection of the string. Replace q Ψ, where Ψ may indicate other properties of the medium that

More information

Joel A. Shapiro January 20, 2011

Joel A. Shapiro January 20, 2011 Joel A. shapiro@physics.rutgers.edu January 20, 2011 Course Information Instructor: Joel Serin 325 5-5500 X 3886, shapiro@physics Book: Jackson: Classical Electrodynamics (3rd Ed.) Web home page: www.physics.rutgers.edu/grad/504

More information

Aerodynamics. High-Lift Devices

Aerodynamics. High-Lift Devices High-Lift Devices Devices to increase the lift coefficient by geometry changes (camber and/or chord) and/or boundary-layer control (avoid flow separation - Flaps, trailing edge devices - Slats, leading

More information

Continuum Mechanics Lecture 5 Ideal fluids

Continuum Mechanics Lecture 5 Ideal fluids Continuum Mechanics Lecture 5 Ideal fluids Prof. http://www.itp.uzh.ch/~teyssier Outline - Helmholtz decomposition - Divergence and curl theorem - Kelvin s circulation theorem - The vorticity equation

More information

Energy during a burst of deceleration

Energy during a burst of deceleration Problem 1. Energy during a burst of deceleration A particle of charge e moves at constant velocity, βc, for t < 0. During the short time interval, 0 < t < t its velocity remains in the same direction but

More information

Active Control of Separated Cascade Flow

Active Control of Separated Cascade Flow Chapter 5 Active Control of Separated Cascade Flow In this chapter, the possibility of active control using a synthetic jet applied to an unconventional axial stator-rotor arrangement is investigated.

More information

Computational Aeroacoustics

Computational Aeroacoustics Computational Aeroacoustics Simple Sources and Lighthill s Analogy Gwénaël Gabard Institute of Sound and Vibration Research University of Southampton, UK gabard@soton.ac.uk ISVR, University of Southampton,

More information

e iωt dt and explained why δ(ω) = 0 for ω 0 but δ(0) =. A crucial property of the delta function, however, is that

e iωt dt and explained why δ(ω) = 0 for ω 0 but δ(0) =. A crucial property of the delta function, however, is that Phys 531 Fourier Transforms In this handout, I will go through the derivations of some of the results I gave in class (Lecture 14, 1/11). I won t reintroduce the concepts though, so you ll want to refer

More information

MATH 205C: STATIONARY PHASE LEMMA

MATH 205C: STATIONARY PHASE LEMMA MATH 205C: STATIONARY PHASE LEMMA For ω, consider an integral of the form I(ω) = e iωf(x) u(x) dx, where u Cc (R n ) complex valued, with support in a compact set K, and f C (R n ) real valued. Thus, I(ω)

More information

THE ANALYSIS OF LAMINATE LAY-UP EFFECT ON THE FLUTTER SPEED OF COMPOSITE STABILIZERS

THE ANALYSIS OF LAMINATE LAY-UP EFFECT ON THE FLUTTER SPEED OF COMPOSITE STABILIZERS THE ANALYSIS OF LAMINATE LAY-UP EFFECT ON THE FLUTTER SPEED OF COMPOSITE STABILIZERS Mirko DINULOVIĆ*, Boško RAŠUO* and Branimir KRSTIĆ** * University of Belgrade, Faculty of Mechanical Engineering, **

More information

Asymptotic Behavior of Waves in a Nonuniform Medium

Asymptotic Behavior of Waves in a Nonuniform Medium Available at http://pvamuedu/aam Appl Appl Math ISSN: 1932-9466 Vol 12, Issue 1 June 217, pp 217 229 Applications Applied Mathematics: An International Journal AAM Asymptotic Behavior of Waves in a Nonuniform

More information

AEROSPACE ENGINEERING

AEROSPACE ENGINEERING AEROSPACE ENGINEERING Subject Code: AE Course Structure Sections/Units Topics Section A Engineering Mathematics Topics (Core) 1 Linear Algebra 2 Calculus 3 Differential Equations 1 Fourier Series Topics

More information

44 CHAPTER 3. CAVITY SCATTERING

44 CHAPTER 3. CAVITY SCATTERING 44 CHAPTER 3. CAVITY SCATTERING For the TE polarization case, the incident wave has the electric field parallel to the x 3 -axis, which is also the infinite axis of the aperture. Since both the incident

More information

Chapter 9 Flow over Immersed Bodies

Chapter 9 Flow over Immersed Bodies 57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 1 Chapter 9 Flow over Immersed Bodies Fluid flows are broadly categorized: 1. Internal flows such as ducts/pipes,

More information

Math 241 Final Exam Spring 2013

Math 241 Final Exam Spring 2013 Name: Math 241 Final Exam Spring 213 1 Instructor (circle one): Epstein Hynd Wong Please turn off and put away all electronic devices. You may use both sides of a 3 5 card for handwritten notes while you

More information

Problem 1. Part a. Part b. Wayne Witzke ProblemSet #4 PHY ! iθ7 7! Complex numbers, circular, and hyperbolic functions. = r (cos θ + i sin θ)

Problem 1. Part a. Part b. Wayne Witzke ProblemSet #4 PHY ! iθ7 7! Complex numbers, circular, and hyperbolic functions. = r (cos θ + i sin θ) Problem Part a Complex numbers, circular, and hyperbolic functions. Use the power series of e x, sin (θ), cos (θ) to prove Euler s identity e iθ cos θ + i sin θ. The power series of e x, sin (θ), cos (θ)

More information

Linear and Nonlinear Oscillators (Lecture 2)

Linear and Nonlinear Oscillators (Lecture 2) Linear and Nonlinear Oscillators (Lecture 2) January 25, 2016 7/441 Lecture outline A simple model of a linear oscillator lies in the foundation of many physical phenomena in accelerator dynamics. A typical

More information

On the Dynamics of Suspension Bridge Decks with Wind-induced Second-order Effects

On the Dynamics of Suspension Bridge Decks with Wind-induced Second-order Effects MMPS 015 Convegno Modelli Matematici per Ponti Sospesi Politecnico di Torino Dipartimento di Scienze Matematiche 17-18 Settembre 015 On the Dynamics of Suspension Bridge Decks with Wind-induced Second-order

More information

Contents. I Introduction 1. Preface. xiii

Contents. I Introduction 1. Preface. xiii Contents Preface xiii I Introduction 1 1 Continuous matter 3 1.1 Molecules................................ 4 1.2 The continuum approximation.................... 6 1.3 Newtonian mechanics.........................

More information

Numerical Solutions to Partial Differential Equations

Numerical Solutions to Partial Differential Equations Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Numerical Methods for Partial Differential Equations Finite Difference Methods

More information

Iterative Learning Control for Smart Rotors in Wind turbines First Results

Iterative Learning Control for Smart Rotors in Wind turbines First Results Iterative Learning Control for Smart Rotors in Wind turbines First Results Owen Tutty 1, Mark Blackwell 2, Eric Rogers 3, Richard Sandberg 1 1 Engineering and the Environment University of Southampton

More information

FUNDAMENTALS OF AERODYNAMICS

FUNDAMENTALS OF AERODYNAMICS *A \ FUNDAMENTALS OF AERODYNAMICS Second Edition John D. Anderson, Jr. Professor of Aerospace Engineering University of Maryland H ' McGraw-Hill, Inc. New York St. Louis San Francisco Auckland Bogota Caracas

More information

Exercise 9: Model of a Submarine

Exercise 9: Model of a Submarine Fluid Mechanics, SG4, HT3 October 4, 3 Eample : Submarine Eercise 9: Model of a Submarine The flow around a submarine moving at a velocity V can be described by the flow caused by a source and a sink with

More information

WAVE PROPAGATION IN ONE- DIMENSIONAL STRUCTURES. Lecture Notes for Day 2

WAVE PROPAGATION IN ONE- DIMENSIONAL STRUCTURES. Lecture Notes for Day 2 WAVE PROPAGATION IN ONE- DIMENSIONAL STRUCTURES Lecture Notes for Day S. V. SOROKIN Department of Mechanical and Manufacturing Engineering, Aalborg University . BASICS OF THE THEORY OF WAVE PROPAGATION

More information

Identification Methods for Structural Systems. Prof. Dr. Eleni Chatzi System Stability - 26 March, 2014

Identification Methods for Structural Systems. Prof. Dr. Eleni Chatzi System Stability - 26 March, 2014 Prof. Dr. Eleni Chatzi System Stability - 26 March, 24 Fundamentals Overview System Stability Assume given a dynamic system with input u(t) and output x(t). The stability property of a dynamic system can

More information

Turbulent spectra generated by singularities

Turbulent spectra generated by singularities Turbulent spectra generated by singularities p. Turbulent spectra generated by singularities E.A. Kuznetsov Landau Institute for Theoretical Physics, Moscow Russia In collaboration with: V. Naulin A.H.

More information

VORTEX LATTICE METHODS FOR HYDROFOILS

VORTEX LATTICE METHODS FOR HYDROFOILS VORTEX LATTICE METHODS FOR HYDROFOILS Alistair Fitt and Neville Fowkes Study group participants A. Fitt, N. Fowkes, D.P. Mason, Eric Newby, E. Eneyew, P, Ngabonziza Industry representative Gerrie Thiart

More information

Traveling Harmonic Waves

Traveling Harmonic Waves Traveling Harmonic Waves 6 January 2016 PHYC 1290 Department of Physics and Atmospheric Science Functional Form for Traveling Waves We can show that traveling waves whose shape does not change with time

More information

Week 2 Notes, Math 865, Tanveer

Week 2 Notes, Math 865, Tanveer Week 2 Notes, Math 865, Tanveer 1. Incompressible constant density equations in different forms Recall we derived the Navier-Stokes equation for incompressible constant density, i.e. homogeneous flows:

More information

Numerical Simulation of Unsteady Aerodynamic Coefficients for Wing Moving Near Ground

Numerical Simulation of Unsteady Aerodynamic Coefficients for Wing Moving Near Ground ISSN -6 International Journal of Advances in Computer Science and Technology (IJACST), Vol., No., Pages : -7 Special Issue of ICCEeT - Held during -5 November, Dubai Numerical Simulation of Unsteady Aerodynamic

More information

The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d

The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d Solving ODEs using Fourier Transforms The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d 2 dx + q f (x) R(x)

More information