Outer Inviscid Flow Research Articles

Improved order of magnitude estimate of rotational effects on HAWTs

This work proposes an improved order of magnitude estimate methodology of rotational effects on wind turbines, by applying the partial pressure field concept to the case of rotating wind turbine blades, which allows isolating acceleration that rises as a result of the rotation of the blades; then an order of magnitude scaling is performed to estimate the partial pressure due to rotation for both fully attached flow and massively separated flow. In the case of fully attached flow, it was found that rotational effects are negligible; while in the case of massively separated flow, rotational effects were found to be important. The pressure reduction was found to scale linearly with the aspect ratio inside the separated boundary layer, while it scales quadratically with the aspect ratio in the outer inviscid flow. A new estimate of the Rossby number is proposed, which is found to be the inverse of the pressure reduction coefficient estimate in the outer inviscid flow. Then, a semi-empirical model of the pressure reduction is proposed based on the order of magnitude estimate in the outer inviscid flow. The proposed semi-empirical model is then compared against the reference data of three wind turbines, namely, MEXICO, NREL phase VI, and AVATAR 10 MW rotors; along with the already known centrifugal pumping model. The results showed that the centrifugal pumping alone, inside the separated boundary layer, is not sufficient to produce the large force augmentation observed in experimental data. In contrast, the improved semi-empirical model provided a good agreement with experimental data, thanks to the quadratic scaling of the aspect ratio. And, the new estimate of the Rossby number proved to be advantageous over the existing Rossby number estimate in terms of the dependency on the angle of attack. Furthermore, the tangential and drag force coefficients were found to be strongly dependent on the airfoil type and its corresponding pressure shape. In contrast, the normal and lift force coefficients were found to be less dependent on the airfoil type. The proposed semi-empirical model is still strongly empirical, future work needs to account for the complex effect of airfoil type and its consequent interaction with the outer inviscid flow through the boundary layer.

Spontaneous radiation of sound by instability of a highly cooled hypersonic boundary layer

The linear stability analysis predicts that the Mack second mode propagating in the boundary layer on a sufficiently cold plate can radiate acoustic waves into the outer inviscid flow. This effect, which is called as a spontaneous radiation (or emission) of sound, is associated with synchronization of the second mode with slow acoustic waves of the continuous spectrum. The theoretical predictions are confirmed by direct numerical simulations of wave trains and wave packets propagating in the boundary layer on a flat plate at free-stream Mach number 6 and wall-to-edge temperature ratio $T_{w}/T_{e}=0.5$. A non-uniform distribution of the wave packet components and the interference between the radiated acoustic waves result in an intricate pattern of the outer acoustic field. The spontaneous radiation of sound, in turn, strongly affects the wave packet in the boundary layer causing its elongation and modulation. This phenomenon may alter the downstream development of instability and delay the transition onset.

Three-dimensional boundary layers with short spanwise scales

AbstractWe investigate three-dimensional (laminar) boundary layers that include a spanwise scale comparable to the boundary-layer thickness. A forcing of short spanwise scales requires viscous dissipation to be retained in the two-dimensional cross-section, perpendicular to the external flow direction, and in this respect the flows are related to previous work on corner boundary layers. We use two examples to highlight the main features of this category of boundary layer: (i) a flat plate of narrow (spanwise) width, and (ii) a narrow (spanwise) gap cut into an otherwise infinite flat plate; in both cases the plate is aligned with a uniform oncoming stream. We find that a novel feature arises in connection with the external flow; the presence of a narrow gap/plate (or indeed any comparable short-scale feature of long streamwise extent) necessarily modifies the streamwise mass flux in that vicinity, which in turn induces an associated boundary-layer transpiration on the same short spanwise length scale. This (short-scale) transpiration region leads to a half-line-source/sink correction to the outer inviscid, irrotational flow. Crucially, the volumetric flux associated with this line-source/sink must be explicitly included as part of the computational procedure for the leading-order boundary layer, and as such there is a weak interaction between the outer (inviscid) flow and the boundary layer. This is a generic feature of boundary layers that are forced through the presence of short-scale spanwise variations.

Analysis of the Airfoil Stall With a Modification of Viscous-Inviscid Interaction Concept

A modification of the viscous-inviscid interaction concept with the employment of coupled vortices around the airfoil wake is introduced for analyzing the airfoil stall. The analyzed flow includes the laminar boundary layers, laminar separation bubble, laminar-turbulent transition zone, turbulent boundary layers, turbulent separation zone, wake, and outer inviscid flow. Integral methods are employed for the boundary layers. The boundaries of separation zones are analyzed as free surfaces, however, their lengths and shapes depend on the Reynolds number. The described modification is validated by a comparison of the numerical results with the previously published experimental data for various airfoils and Reynolds numbers at low Mach numbers. This modification achieves a reasonably good agreement of the computed lift and moment coefficients with their measured values.

Interactive planar multi-blade flows with a global angle of attack

Planar flow past multiple successive blades and wakes is studied for locally non-symmetric blades aligned with a global angle of attack to the oncoming freestream. The typical blade lies relatively near the centreline of the oncoming wake from the preceding blade but is shifted due to the global attack angle. The central flow structure is of a periodic, pressure interactive boundary-layer over each blade and wake, embedded within a slowly growing Blasius-type flow. This is coupled through the inviscid outer flow via a pressure–displacement relationship. Differing amendments to this structure occur as the global angle of attack is increased in magnitude. The local and global non-symmetries require a leading-edge jump in pressure and velocities in the inner boundary-layer to ensure that the Kutta condition is satisfied at the trailing edge. The flow structure is described for a range of angles of attack, and numerical results are presented for the two cases of most interest where the innermost periodic layer is directly influenced by the global features. Studies are restricted to the condensed limit here. Separation is found to occur and can be caused by both local and global influences.

The Basis of an Industry-standard Aerodynamic Computation and its Theoretical Basis with Inferences and Applications

“The PNS method is in very widespread use; indeed, it forms the basis of an industry-standard computer program, which is used by virtually all major aerodynamics laboratories and companies.”(John D. Anderson Jr. American aerodynamicist 2006). Several PNS Eqs. With different slightly viscous terms and different name were suggested between the late 1960s and the late 1970s, in which the simplified NS Eqs. suggested by Z.Gao in 1967 was called by F.G.Zhuang(Academician, China) etal as first PNS. In early studies of PNS, it is not very clear what kind of basic flow is described by PNS Eqs., there is no basic fluid theory corresponding to PNS. The author presented a PNS's fluid basic theory: viscous/inviscid interacting shear flow (ISF) theory (including laminar, perturbed and turbulent flow theories). ISF consists of viscous shear layer and neighboring outer inviscid flow. Main points of ISF theory and inferences are as follows. Both the flow law and mathematical definition of ISF's viscous layer and ISF's governing Eqs. written in ISF's optimal coordinates, that is a fitted dividing flow surface orthogonal coordinates, are presented. The evolution laws of physical scales of ISF's viscous layer flow are solved, which do generate small scale structure and then do cause local sudden increase of several physical quantities(such as heat flux etc.). The PNS Eqs. can be simplified further on the dividing flow surface and the resultant Eqs. are called dividing flow surface criteria, whose two special cases are wall-surface criteria (WSC)for solutions of NS- and Euler Eqs. NS numerical solutions ought to satisfy WSC, which are proved numerically by some computations. People can obtain NS grid independent solution by operation of NS numerical solutions satisfying progressively the WSC. Especially, this criteria can evaluate the departure of NS numerical solutions from NS true solution by just one time NS calculation on one set of grid and are theoretical methods evaluating creditability of NS numerical solutions, and based on that some people suggest that the WSC method may be used as an ingenious substitute for the grid convergence analysis method in common use and that the WSC method and ISF theory are called Gao's criteria method and Gao's ISF theory, respectively. Some applications of ISF theory and its inferences to CFD are fruitful and creativeness and then research on this aspect is worth deepening and widening.

Slip flow on a body of revolution

This work investigates the boundary layer development and resulting net frictional drag along a general blunt-nosed body of revolution in uniform slip flow. The curvilinear boundary layer equations are subjected to a Navier slip condition near the surface. For motion near the nose, a similarity-type solution is obtained as a double series, comprising a small velocity slip parameter, from which the non-dimensional drag coefficient is obtained. In addition, the displacement effect of the boundary layer on the outer inviscid flow is determined and interpreted by means of a dimensionless parameter. The general analysis is then applied to the special case of the Rankine half-body. Considering laminar flow along the nose and downstream regions, results are presented for prescribed slip parameters and Reynolds numbers with particular emphasis on the influence of slip on the flow.

High Reynolds number Navier–Stokes solutions and boundary layer separation induced by a rectilinear vortex

We compute the solutions of Prandtl’s and Navier–Stokes equations for the two dimensional flow induced by a rectilinear vortex interacting with a boundary in the half plane. For this initial datum Prandtl’s equation develops, in a finite time, a separation singularity. We investigate the different stages of unsteady separation for Navier–Stokes solution at different Reynolds numbers Re = 10 3–10 5, and we show the presence of a large-scale interaction between the viscous boundary layer and the inviscid outer flow. We also see a subsequent stage, characterized by the presence of a small-scale interaction, which is visible only for moderate-high Re numbers Re = 10 4–10 5. We also investigate the asymptotic validity of boundary layer theory by comparing Prandtl’s solution to Navier–Stokes solutions during the various stages of unsteady separation.

The wave-induced boundary layer under long internal waves

The boundary layer formed under the footprint of an internal solitary wave is studied by numerical simulation for waves of depression in a two-layer model of the density stratification. The inviscid outer flow, in the perspective of boundary-layer theory, is based on an exact solution for the long wave-phase speed, yielding a family of fully nonlinear solitary wave solutions of the extended Korteweg–de Vries equation. The wave-induced boundary layer corresponding to this outer flow is then studied by means of simulation employing the Reynolds-averaged Navier–Stokes (RANS) formulation coupled with a turbulence closure model validated for wall-bounded flows. Boundary-layer characteristics are computed for an extensive range of environmental conditions and wave amplitudes. Boundary-layer transition, identified by monitoring the eddy viscosity, is correlated in terms of a boundary-layer Reynolds number. The frictional drag is evaluated for laminar, transitional, and turbulent cases, and correlations are presented for the friction coefficient plus relevant measures of the boundary-layer thickness.

Cavity detachment on a hydrofoil with the inclusion of surface tension effects

The non-linear problem of cavity flow past a hydrofoil is considered with taking into account fluid viscosity in the cavity closure region and surface tension, which affect the cavity detachment. The theoretical model is based on the concept of viscous–inviscid interaction between the outer inviscid cavity flow and the inner turbulent separated flow downstream of the cavity. The outer inviscid flow is solved by constructing the complex flow potential, and the wake model is based on the method of integral relationships for separated turbulent flows. The obtained numerical results are compared with experimental data.

Investigation of the flows in axisymmetric and plane divergent channels with preseparation boundary layers on their walls

The method of matched asymptotic expansions is used to develop a procedure for constructing separationless plane and axisymmetric diffusers for incompressible high-Reynolds-number flows with a given pressure distribution over the diffuser surface and minimum friction losses. The problem for the region of interaction between the boundary layer and the inviscid outer flow is numerically solved and the effect of variation of the on-design flow parameters in the entry section on the diffuser flow parameters is studied.

ANALYSIS OF THREE WING SECTIONS

Three wing sections FX66-S-196VI, E603 and AH82–150A were analyzed. Measured data of these wing sections are published. First wing section was measured in Stuttgart University and in Delft University of Technology, when other two wing sections were measured in Stuttgart University. These wing sections have different behavior in the region of maximum lift. FX66‐S‐196VI wing section has sharp drop in lift. The stall of the second and third wing sections is smooth, though different. All wing sections are affected by laminar separation bubbles. The calculations were performed using three codes: Eppler Program System, XFOIL and RFOIL. Eppler's code uses non‐interacted inviscid plus boundary layer method. Influence of separation is estimated using empirical correction in this method. XFOIL code of Mark Drela, MIT uses interacted zonal viscous/inviscid method. The wall transpiration model in this code approximates the displacement effect on the outer inviscid flow. RFOIL is a modification of XFOIL code for application in wind turbines performed at Delft University of Technology. The code's prediction of the airfoil performance around the two dimensional maximum lift was enhanced. The comparison of calculated and measured data is presented and analyzed. Santrauka Tyrimuose analizuotos profiliu FX 66-S-196 V1, E603 ir AH82–150A charakteristiku teorines reikšmes, apskaičiuotos XFOIL, RFOIL ir PROFIL05 programomis. Gautos teorines reikšmes palygintos su jau atliktu eksperimentiniu tyrinejimu rezultatais. Pirmas profilis buvo tyrinejamas Delft technologijos universitete (Olandija) ir Štutgarto universitete (Vokietija), like du ‐ Štutgarto universitete. Visi profiliai turi skirtingas maksimalios keliamosios jegos dalis. Profilio FX 66‐S‐196 V1 keliamoji jega mažeja staiga. Kitu profiliu keliamoji jega kinta tolygiai, tačiau skirtingai. Visu profiliu ucharakteristikas itakoja laminarinis atsiskyrimo burbulas.

A Structured Mesh Euler and Interactive Boundary Layer Method for Wing/Body Configurations

To compute transonic flows over a complex 3D aircraft configuration, a viscous/inviscid interaction method is developed by coupling an integral boundary-layer solver with an Eluer solver in a “semi-inverse” manner. For the turbulent boundary-layer, an integral method using Green's lag equation is coupled with the outer inviscid flow. A blowing velocity approach is used to simulate the displacement effects of the boundary layer. To predict the aerodynamic drag, it is developed a numerical technique called far-field method that is based on the momentum theorem, in which the total drag is divided into three component drags, i.e. viscous, induced and wave-formed. Consequently, it can provide more physical insight into the drag sources than the often-used surface integral technique. The drag decomposition can be achieved with help of the second law of thermodynamics, which implies that entropy increases and total pressure decreases only across shock wave along a streamline of an inviscid non-isentropic flow. This method has been applied to the DLR-F4 wing/body configuration showing results in good agreement with the wind tunnel data.

Study of laminar separation in the vicinity of the point of piecewise-constant pressure distribution over the wall

A family of two-dimensional divergent channels with piecewise-constant velocity and pressure distributions over the wall is considered. The method of matched asymptotic expansions is applied to study the two-dimensional viscous incompressible flow at high but subcritical Reynolds numbers in the vicinity of a pressure jump point on the channel wall. It is shown that if the pressure difference is of the order O(Re−1/4), then in the vicinity of this point a classical region of interaction between the viscous boundary layer on the wall and the outer inviscid flow occurs. The problem formulated for the interaction region is solved numerically. The asymptotic values of the pressure difference corresponding to separationless flow are determined and the separation flow patterns are constructed.

Laminar separation bubbles from a high Reynolds‐number asymptotic point of view

AbstractClassical boundary layer theory proved to lose its validity if flow separation is encountered. However, introduction of the triple deck concept – i.e. allowing for the displacement effect of the viscous boundary layer to interact locally with the induced pressure (disturbances) in the outer inviscid flow regime – led to a successful description of ‘short’ laminar separation bubbles in various fundamental flow problems. Here we want to focus on situations which share the following distinctive feature: two‐dimensional steady state solutions of the underlying interaction equations exist in a limited range of values of a parameter controlling the conditions leading to separation only. Furthermore, these solutions turn out to be non‐unique in the vicinity of the limiting critical value of this parameter. Typical examples include leading edge separation, smooth backward facing step flow, channel flow with suction (cases of so‐called marginal separation) and subsonic expansion ramp flow, asymmetric trailing edge separation (classical triple deck flows). More recent investigations which take into account (local) unsteady, three‐dimensional and flow control device effects and concentrate on near critical flows indicate that (i) exceeding the critical value of the controlling parameter is associated with self‐sustained vortex shedding (time periodic bubble ‘bursting’ even in the absence of external disturbances), (ii) coherent structure formation caused by bubble bursting events turns out to be essentially self‐similar, (iii) the high sensitivity of the separated flow region with respect to external disturbances allows for effective application of (optimized) flow control devices in order to force or delay the transition process to turbulent boundary layer flow. The aim of the presentation is to deliver insight into the power of modern asymptotic analysis. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Method for solving the equations describing the interaction of a three-dimensional boundary layer with an outer inviscid flow

A new method is developed for solving the three-dimensional time-independent equations describing the interaction of a laminar boundary layer with an outer inviscid flow. The method also applies to the interaction of plane flows. By applying the method, the problem of the three-dimensional viscous supersonic gas flow over a roughness element (a hump and a cavity) is solved for the first time within the framework of the classical triple-deck theory. The asymptotic height of the roughness element corresponding to the nonseparated flow is determined, and separated flow patterns are constructed.

Numerical simulation of steady compressible flows using a zonal formulation. Part II: viscous flows

Viscous flow simulations are usually based on the Navier–Stokes equations representing the balance of mass, momentum, and energy. For many high Reynolds number flows, the viscous effects are only limited to small regions in the neighborhood of solid surfaces and in the wakes. We present here a zonal formulation with a reduced system of equations in the outer inviscid flow region. As in part I of this work, the velocity components are calculated from a generalized form of the Cauchy–Riemann equations with non-vanishing vorticity only in the inner region, where the governing equations including the viscous terms are solved. The viscous effects are transferred to the Cauchy–Riemann equations through a forcing function (vorticity), and the process is repeated until convergence. Preliminary results are presented and compared to standard Navier–Stokes calculations for two and three dimensional flow problems.

Reduced-Order Modeling of Unsteady Viscous Flow in a Compressor Cascade

A simultaneously coupled viscous-inviscid interaction (VII) analysis is used to model the unsteady viscous separated flow through a subsonic compressor. The inner viscous flow around the airfoil and in the wake is modeled using a finite difference discretization of the boundary-layer equations and a one-equation turbulence transport model. The outer inviscid flow is modeled using a variational finite element discretization of the compressible full potential equation. The viscous and inviscid regions are simultaneously coupled using a injection type boundary condition along the airfoil and wake. The resulting nonlinear unsteady equations are linearized about the nonlinear steady flow to obtain a set of linear equations that describe the unsteady small-disturbance behavior of the viscous flow through the cascade. The discretized small-disturbance VII equations are used to form a generalized, quadratic, non-Hermitian eigenvalue problem that describes the eigenmodes (natural modes) and eigenvalues (natural frequencies) of fluid motion about the cascade. Using a Lanczos algorithm, the eigeninformation is computed efficiently for various steady flow inflow angles and unsteady interblade phase angles

Concerning unsteady boundary layers formulated in a non-inertial reference frame

Formulation of incompressible boundary layers over generally accelerating two-dimensional bodies is addressed from the viewpoint of a non-inertial reference frame. Starting from the Navier-Stokes equations written in terms of a set of arbitrarily moving coordinates, the boundary-layer equations with reference to a body-fixed frame are derived that shows the equations retain essentially the same forms as in inertial cases. Expression of the outer inviscid flow needed to complete the description of the boundary layer around an arbitrarily accelerating body can be obtained by appealing to the concept of the ‘boundary stream function’ and techniques from the complex theory. As an example, the formulation of boundary layers over thin elliptic cylinders undergoing pitching/plunging motions is presented. A comparison of the inviscid-flow responses to the two motions is also made, and the validity of the classical linearized aerodynamic theory is discussed.

Topological interpretation of the surface flow visualization of conical viscous/inviscid interactions

The asymptotic flow structure is considered for a viscous–inviscid conical interaction, in particular that between a swept shock wave and a boundary layer. A flow model is devised based on the three-layer interaction concept. Assuming conicity of the inviscid flow regions, a viscous layer structure is established that is compatible with the inviscid outer flow, and which produces a geometrically conical surface flow pattern. This result is obtained from a dimensional analysis, which reveals similarity of the viscous layer in cross-flow planes at different radial distances from the conical origin. The results of this analysis provide a tool for the quantitative interpretation of surface flow visualizations in terms of the related topological structure of the flow in the cross-flow plane. This method is illustrated by application to the surface flow visualization of a Mach 3 shock-wave/boundary-layer interaction.