Far-Field Analysis of the Aerodynamic Force by Lamb Vector Integrals

Marongiu, Claudio; Tognaccini, Renato

doi:10.2514/1.j050326

Cited by 52 publications

(20 citation statements)

References 20 publications

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“…Different approaches have been used to deal with the troublesome pressure term, which leads to various force expressions. [17][18][19][20][21][22][23] However, for a general control surface, eliminating the pressure term usually results in more complicated expressions in which the physical meanings and relative contributions of some terms cannot be easily elucidated. By introducing an auxiliary velocity potential satisfying suitable boundary conditions and projecting the NS equations on the gradient of the potential, Quartapelle and Napolitano, 28 Howe, 18 and Chang 17 were able to extract the explicit pressure force from the pressure term and decompose the force into several terms whose physical meanings become clear.…”

Section: A Fluid-mechanic Forcementioning

confidence: 99%

A lift formula applied to low-Reynolds-number unsteady flows

Wang

Zhang

et al. 2013

Physics of Fluids

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A lift formula for a wing in a rectangular control volume is given in a very simple and physically lucid form, providing a rational foundation for calculation of the lift of a flapping wing in highly unsteady and separated flows at low Reynolds numbers. Direct numerical simulations on the stationary and flapping two-dimensional flat plate and rectangular flat-plate wing are conducted to assess the accuracy of the lift formula along with the classical Kutta-Joukowski theorem. In particular, the Lamb vector integral for the vortex force and the acceleration term of fluid for the unsteady inertial effect are evaluated as the main contributions to the unsteady lift generation of a flapping wing. C 2013 AIP Publishing LLC. [http://dx

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Section: A Fluid-mechanic Forcementioning

confidence: 99%

A lift formula applied to low-Reynolds-number unsteady flows

Wang

Zhang

et al. 2013

Physics of Fluids

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“…The general force expressions have been extensively discussed by Saffman [16] in the framework of inviscid flows. From the NavierStokes (NS) equations, various force expressions have been derived depending on how to transform the pressure term to the velocity-related quantities [17][18][19][20][21][22][23][24][25]. In particular, Noca et al [22] gave the force expressions called "impulse equation", "momentum equation", and "flux equation".…”

Section: Nomenclaturesmentioning

confidence: 99%

“…The general force expressions have been derived from the NS equations in [17][18][19][20][21][22][23][24][25]. Here, the simple lift formula and the typical lift formulas given by Noca et al [22] and Wu et al [24] are summarized.…”

Section: Lift Formulasmentioning

confidence: 99%

Evaluation of Lift Formulas Applied to Low-Reynolds-Number Unsteady Flows

Wang

Zhang

et al. 2015

AIAA Journal

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Different lift decompositions into the elemental terms are compared based on direct numerical simulations of a flapping flat plate and a flapping rectangular wing at low-Reynolds-number flows. The simple lift formula is given as a useful approximate form that has the vortex force and local acceleration terms. The accuracy of the simple lift formula in lift estimation is quantitatively evaluated in comparison with the general force formulas based on the fully resolved two-and three-dimensional unsteady velocity fields and the planar velocity fields at several spanwise locations in simulated particle-image-velocimetry measurements. In addition, the mathematical connections between the different force formulas are discussed.

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“…A breakdown of the drag for compressible flows was suggested by Mele and Tognaccini [7]. Finally, Marongiu and Tognaccini [8] proposed a breakdown in the unsteady case. These formulations may be promising for a drag breakdown in the unsteady case, since the physics involved is richer, and should allow definition of the induced component directly rather than by default.…”

mentioning

confidence: 93%

Improved Unsteady Far-Field Drag Breakdown Method and Application to Complex Cases

2016

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J054756Far-field drag extraction has the advantage over near-field integration of providing a phenomenological breakdown of drag. A decomposition into components linked to shock waves, viscous interactions, and lift-induced vortices is straightforward for steady flows. A formulation based on thermodynamic considerations is used at ONERA-The French Aerospace Lab and by its partners, but it is restricted to steady cases. A generalization to unsteady flows of the Van der Vooren formulation has been developed and tested on three unsteady cases previously. The proposed method allowed the breakdown of drag into the three usual components only; however, the induced drag coefficient remained ill-defined. This unsteady formulation is here modified to better express the induced drag. A new drag component is identified as a propagation and acoustics contribution. The new formulation is then applied to complex cases: two-dimensional and three-dimensional pitching cases, and an OAT15A profile subject to buffet simulated by zonal detached-eddy simulation computations.new propagation and acoustics drag D sp = spurious drag D ui = unsteady induced drag D v = viscous drag D vw = profile drag D w = wave drag H = stagnation enthalpy; h q 2 ∕2 h = enthalpy i = freestream direction vector k = reduced frequency for the pitching velocity ω; 2kM ∞ ∕c M = Mach number n = normal vector pointing outside the flow domain p = static pressure p i = stagnation pressure p1 γ − 1∕2M 2 γ∕γ−1 q = velocity vector Re = Reynolds number r = gas constant S a = surface of the body S cd = complementary downstream wake plane; S d \ S wd ∪ S vd S d = downstream wake plane S e = outer surface of the fluid volume S v = surface for the integration of viscous drag S vd= downstream wake plane of the streamtube enclosing the body and its boundary layer S w = surface for the integration of wave drag S wd = downstream wake plane of the streamtube enclosing the shock s = entropy T = temperature t = time u, v, w = velocity components in an inertial reference frame u irr = axial velocity under irreversible flow assumptions= volume enclosed within S v V w = volume enclosed within S w V wd = volume downstream of V w α = angle of attack γ = ratio of specific heats ρ = density τ = deviatoric stress tensor τ x = longitudinal stress vector; τ · i Subscript ∞ = freestream state

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Far-Field Analysis of the Aerodynamic Force by Lamb Vector Integrals

Cited by 52 publications

References 20 publications

A lift formula applied to low-Reynolds-number unsteady flows

A lift formula applied to low-Reynolds-number unsteady flows

Evaluation of Lift Formulas Applied to Low-Reynolds-Number Unsteady Flows

Improved Unsteady Far-Field Drag Breakdown Method and Application to Complex Cases

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