scholarly journals Note on the Physical Basis of the Kutta Condition in Unsteady Two-Dimensional Panel Methods

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
M. La Mantia ◽  
P. Dabnichki
Keyword(s):  
Pressure Difference ◽  
Energy Requirements ◽  
Aquatic Species ◽  
Biological Applications ◽  
Velocity Difference ◽  
Trailing Edge ◽  
Kutta Condition ◽  
Wing Motion ◽  
Panel Methods ◽  
New Formulation

Force generation in avian and aquatic species is of considerable interest for possible engineering applications. The aim of this work is to highlight the theoretical and physical foundations of a new formulation of the unsteady Kutta condition, which postulates a finite pressure difference at the trailing edge of the foil. The condition, necessary to obtain a unique solution and derived from the unsteady Bernoulli equation, implies that the energy supplied for the wing motion generates trailing-edge vortices and their overall effect, which depends on the motion initial parameters, is a jet of fluid that propels the wing. The postulated pressure difference (the value of which should be experimentally obtained) models the trailing-edge velocity difference that generates the thrust-producing jet. Although the average thrust values computed by the proposed method are comparable to those calculated by assuming null pressure difference at the trailing edge, the latter (commonly used) approach is less physically meaningful than the present one, as there is a singularity at the foil trailing edge. Additionally, in biological applications, that is, for autonomous flapping, the differences ought to be more significant, as the corresponding energy requirements should be substantially altered, compared to the studied oscillatory motions.

Part I: Some experiences on simple panel methods applied to attached and separated flows

1987 ◽  
Vol 91 (908) ◽  
pp. 350-359 ◽  
Author(s):  
H. B. Tou ◽  
G. J. Hancock
Keyword(s):  
Piecewise Linear ◽  
Separated Flows ◽  
Surface Singularity ◽  
List Type ◽  
Trailing Edge ◽  
Kutta Condition ◽  
First Order ◽  
Panel Methods ◽  
Total Head ◽  
Order Surface

Summary Simple first order surface singularity methods based on: (i) Smith and Hess uniform source panels plus a uniform vorticity around aerofoil profile, (ii) piecewise linear vorticity around aerofoil profile, with different assumption for Kutta condition, have been applied to attached flows and separated flows past an aerofoil/spoiler configuration, assuming an inviscid model. For separated flows, piecewise linear vorticity methods give reasonable results as long as small panel elements are taken in the region of the separations at the spoiler tip and aerofoil trailing edge. The Smith and Hess method gives results which do not agree too closely with the vorticity methods. There is doubt concerning uniqueness. Results have been compared using two different wake models; in one, the total head inside the wake is taken to be uniform, in the second, the static pressures along the separation streamlines are taken to be uniform. There appears to be a difference of about 5% in CL. It is not known why.

Vorticity and Kutta Condition for Unsteady Multienergy Flows

1969 ◽  
Vol 36 (3) ◽  
pp. 608-613 ◽  
Author(s):  
J. P. Giesing
Keyword(s):  
Vortex Shedding ◽  
Pressure Difference ◽  
Total Pressure ◽  
Vortex Sheet ◽  
Rate Of Change ◽  
The Other ◽  
Numerical Calculations ◽  
Time Rate ◽  
Trailing Edge ◽  
Kutta Condition

The dynamical conditions for vortex shedding in unsteady multienergy flows are given: It is shown that the vorticity shed is composed of an unsteady part, which is proportional to the time rate of change of the circulation, and a steady part, which is proportional to the total-pressure difference across the vortex sheet. The kinematics of vortex shedding are also investigated. It is determined that the vortex sheet is shed parallel to one side of the trailing edge or the other depending on the sense of the shed vorticity. It is further determined that the shedding velocity is equal to one half of the strength of the vorticity at the trailing edge (except for trailing-edge angles of zero). Numerical calculations are presented to illustrate the results.

scholarly journals On the Kutta Condition in Potential Flow over Airfoil

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Farzad Mohebbi ◽  
Mathieu Sellier
Keyword(s):  
Potential Flow ◽  
Trailing Edge ◽  
Airfoil Surface ◽  
Kutta Condition ◽  
Edge Angle ◽  
Panel Methods ◽  
Flow Potential ◽  
Novel Method ◽  
Elliptic Grid Generation ◽  
Fast Procedure

This paper proposes a novel method to implement the Kutta condition in irrotational, inviscid, incompressible flow (potential flow) over an airfoil. In contrast to common practice, this method is not based on the panel method. It is based on a finite difference scheme formulated on a boundary-fitted grid using an O-type elliptic grid generation technique. The proposed algorithm uses a novel and fast procedure to implement the Kutta condition by calculating the stream function over the airfoil surface through the derived expression for the airfoils with both finite trailing edge angle and cusped trailing edge. The results obtained show the excellent agreement with the results from analytical and panel methods thereby confirming the accuracy and correctness of the proposed method.

Rotor Wake Generated Unsteady Aerodynamic Response of a Compressor Stator

1978 ◽  
Vol 100 (4) ◽  
pp. 664-675 ◽  
Author(s):  
S. Fleeter ◽  
R. L. Jay ◽  
W. A. Bennett
Keyword(s):  
Pressure Difference ◽  
Second Harmonic ◽  
Incidence Angle ◽  
Dynamic Pressure ◽  
Pressure Differential ◽  
Phase Lag ◽  
Trailing Edge ◽  
Unsteady Pressure ◽  
Forcing Function ◽  
Reduced Frequency

An experimental investigation was conducted to determine the fluctuating pressure distribution on a stationary vane row, with the primary source of excitation being the wakes from the upstream rotor blades. This was accomplished in a large scale, low speed, single stage research compressor. The forcing function, the velocity defect created by the rotor wakes, was measured with a crossed hot-wire probe. The aerodynamic response on the vanes was measured by means of flush mounted high response dynamic pressure transducers. The dynamic data were analyzed to determine the chordwise distribution of the dynamic pressure coefficient and aerodynamic phase lag as referenced to a transverse gust at the vane leading edge. Vane suction and pressure surface data as well as the pressure difference across the vane were obtained for reduced frequency values ranging from 3.65 to 16.80 and for an incidence angle range of 35.5 deg. The pressure difference data were correlated with a state-of-the-art aerodynamic cascade transverse gust analysis. The correlation was quite good for all reduced frequency values for small values of incidence. For the more negative incidence angle data points, it was shown that a convected wake phenomena not modeled in the analysis existed. Both the first and second harmonic unsteady pressure differential magnitude data decrease in the chordwise direction. The second harmonic magnitude data attains a value very nearly zero at the vane trailing edge transducer location, while the first harmonic data is still finite, albeit small, at this location. That the magnitude of the unsteady pressure differential data approaches zero near to the trailing edge, particularly the second harmonic data which has reduced frequency values to 16.8, is significant in that it reflects upon the validity of the Kutta condition for unsteady flows.

Viscous tails in Hele-Shaw flow

1971 ◽  
Vol 46 (3) ◽  
pp. 569-576
Author(s):  
C. J. Wood
Keyword(s):  
Reynolds Number ◽  
Trailing Edge ◽  
Kutta Condition ◽  
Quantitative Discrepancy ◽  
Edge Flow ◽  
Hele Shaw Cell

An experiment has been performed, using pulsed dye injection on an aerofoil in a Hele-Shaw cell. The purpose was to observe the form of the trailing-edge flow when the Reynolds number was high enough to permit separation and the initiation of a Kutta condition. The experiment provides a successful confirmation of the existence of a ‘viscous tail’ as predicted by Buckmaster (1970) although there is an unexplained quantitative discrepancy.

Airfoil Design in Subcritical and Supercritical Flows

1979 ◽  
Vol 46 (4) ◽  
pp. 761-766 ◽  
Author(s):  
W. C. Chin ◽  
D. P. Rizzetta
Keyword(s):  
Reasonable Agreement ◽  
Relaxation Method ◽  
Supercritical Pressure ◽  
Small Disturbance ◽  
Trailing Edge ◽  
Difference Analysis ◽  
Pressure Distributions ◽  
Basic Ideas ◽  
New Formulation ◽  
Supercritical Flows

The “inverse” or “design” problem in aerodynamics, which solves for the airfoil shape that induces a prescribed chordwise surface pressure subject to additional requirements on trailing edge closure, is considered in the transonic small-disturbance limit. A new formulation for the stream function ψ is suggested which uses well-set Neumann conditions on the chordwise slit, with the degree of closure dictated by a specified jump in ψ across the downstream slit emanating from the trailing edge. The boundary-value problem is solved by a type-dependent relaxation method that automatically generates closed airfoils on convergence. Computed airfoil shapes using subcritical and supercritical pressure distributions obtained from existing finite-difference analysis codes, in the latter case, with and without shockwaves, give results in reasonable agreement with the original specified shapes, and validate the basic ideas.

A B-Spline Higher-Order Panel Method Applied to Two-Dimensional Lifting Problem

2003 ◽  
Vol 47 (04) ◽  
pp. 290-298
Author(s):  
Chang-Sup Lee ◽  
Justin E. Kerwin
Keyword(s):  
Present Method ◽  
Gaussian Quadrature ◽  
Solution Process ◽  
Higher Order ◽  
Panel Method ◽  
Pressure Jump ◽  
Two Dimensional ◽  
Kutta Condition ◽  
B Spline ◽  
Panel Methods

A higher-order panel method based on B-spline representation for both the geometry and the solution is developed for the solution of the flow around two-dimensional lifting bodies. The influence functions due to the normal dipole and the source are separated into the singular and nonsingular parts; then the former is integrated analytically, whereas the latter is integrated using Gaussian quadrature. Through a desingularization process, the accuracy of the present method can be increased without limit to any order by selecting a proper numerical quadrature. A null pressure jump Kutta condition at the trailing edge is found to be effective in stabilizing the solution process and in predicting the correct solution. Numerical experiments indicate that the present method is robust and predicts the pressure distribution around lifting foils with far fewer panels than existing low-order panel methods.

Sound diffraction at a trailing edge

1981 ◽  
Vol 108 ◽  
pp. 443-460 ◽  
Author(s):  
S. W. Rienstra
Keyword(s):  
Vortex Shedding ◽  
Pulse Wave ◽  
Layer Structure ◽  
Harmonic Wave ◽  
Sound Field ◽  
Edge Condition ◽  
Trailing Edge ◽  
Kutta Condition ◽  
High Reynolds ◽  
Sound Diffraction

The diffraction of externally generated sound in a uniformly moving flow at the trailing edge of a semi-infinite flat plate is studied. In particular, the coupling of the sound field to the hydrodynamic field by way of vortex shedding from the edge is considered in detail, both in inviscid and in viscous flow.In the inviscid model the (two-dimensional) diffracted fields of a cylindrical pulse wave, a plane harmonic wave and a plane pulse wave are calculated. The viscous proess of vortex shedding is represented by an appropriate trailing-edge condition. Two specific cases are compared, in one of which the full Kutta condition is applied, and in the other no vortex shedding is permitted. The results show good agreement with Heavens’ (1978) observations from his schlieren photographs, and confirm his conclusions. It is further demonstrated, by an explicit expression, that the sound power absorbed by the wake may be positive or negative, depending on Mach number and source position. So the process of vortex shedding does not necessarily imply an attenuation of the sound.In the viscous model a high-Reynolds-number approximation is constructed, based on a triple-deck boundary-layer structure, matching the harmonic plane wave outer solution to a known incompressible inner solution near the edge, to obtain the viscous correction to the Kutta condition.

A theoretical model for multiply connected wings

1998 ◽  
Vol 9 (6) ◽  
pp. 607-634
Author(s):  
P. BASSANINI ◽  
C. M. CASCIOLA ◽  
M. R. LANCIA ◽  
R. PIVA
Keyword(s):  
Three Dimensional ◽  
Inviscid Flow ◽  
Linear Functional Equation ◽  
Trailing Edge ◽  
Vortex Sheets ◽  
Kutta Condition ◽  
Multiply Connected ◽  
Flow Solution ◽  
Multiplicative Factor

Steady incompressible inviscid flow past a three-dimensional multiconnected (toroidal) aerofoil with a sharp trailing edge TE is considered, adopting for simplicity a linearized analysis of the vortex sheets that collect the released vorticity and form the trailing wake. The main purpose of the paper is to discuss the uniqueness of the bounded flow solution and the role of the eigenfunction. A generic admissible flow velocity u has an unbounded singularity at TE; and the physical flow solution requires the removal of the divergent part of u (the Kutta condition). This process yields a linear functional equation along the trailing edge involving both the normal vorticity ω released into the wake, and the multiplicative factor of the eigenfunction, a1. Uniqueness is then shown to depend upon the topology of the trailing edge. If δTE=[empty ], as, for example, in an annular-aerofoil configuration, both ω and a1 are uniquely determined by the Kutta condition, and the bounded flow u is unique. If δTE≠[empty ], as, for example, in a connected-wing configuration, there is an infinity of bounded flows, parametrized by a1. Numerical results of relevance for these typical configurations are presented to show the different role of the eigenfunction in the two cases.

Correlated unsteady and steady laminar trailing-edge flows

1981 ◽  
Vol 108 ◽  
pp. 171-183 ◽  
Author(s):  
S. N. Brown ◽  
H. K. Cheng
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Small Amplitude ◽  
Frequency Parameter ◽  
Trailing Edge ◽  
Kutta Condition ◽  
Sinusoidal Oscillations ◽  
Steady Laminar ◽  
Uniform Stream

The incompressible laminar flow in the neighbourhood of the trailing edge of an aerofoil undergoing sinusoidal oscillations of small amplitude in a uniform stream is described in the limit as the Reynolds number R tends to infinity. It is shown that if the frequency parameter is of any order less than R¼ the viscous correction to the Kutta condition and hence to the lift and moment may be determined from the results for the steady case. Justification of this correlation requires discussion of the flow in an additional region not encountered in previous studies.

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