scholarly journals Analysis of complex singularities in high-Reynolds-number Navier–Stokes solutions

2014 ◽  
Vol 747 ◽  
pp. 381-421 ◽  
Author(s):  
F. Gargano ◽  
M. Sammartino ◽  
V. Sciacca ◽  
K. W. Cassel
Keyword(s):  
Boundary Layer ◽  
Large Scale ◽  
Reynolds Numbers ◽  
Singularity Analysis ◽  
Separation Process ◽  
Navier Stokes ◽  
Complex Singularity ◽  
Scale Interaction ◽  
Complex Singularities ◽  
Prandtl Boundary Layer

AbstractNumerical solutions of the laminar Prandtl boundary-layer and Navier–Stokes equations are considered for the case of the two-dimensional uniform flow past an impulsively-started circular cylinder. The various viscous–inviscid interactions that occur during the unsteady separation process are investigated by applying complex singularity analysis to the wall shear and streamwise velocity component of the two solutions. This is carried out using two different methodologies, namely a singularity-tracking method and the Padé approximation. It is shown how the van Dommelen and Shen singularity that occurs in solutions of the Prandtl boundary-layer equations evolves in the complex plane before leading to a separation singularity in finite time. Navier–Stokes solutions, computed at different Reynolds numbers in the range$10^3 \leq Re \leq 10^5$, are characterized by the presence of various complex singularities that can be related to different physical interactions acting over multiple spatial scales. The first interaction developing in the separation process is large-scale interaction that is visible for all the Reynolds numbers considered, and it signals the first relevant differences between the Prandtl and Navier–Stokes solutions. For$Re\geq O(10^4)$, a small-scale interaction follows the large-scale interaction. The onset of these interactions is related to the characteristic changes of the streamwise pressure gradient on the circular cylinder. Even if these interactions physically differ from that prescribed by the Prandtl solution, and they set a possible limit on the comparison of Prandtl solutions with Navier–Stokes solutions, it is shown how the asymptotic validity of boundary-layer theory is strongly supported by the results that have been obtained through the complex singularity analysis.

Navier–Stokes solutions of unsteady separation induced by a vortex

2002 ◽  
Vol 465 ◽  
pp. 99-130 ◽  
Author(s):  
A. V. OBABKO ◽  
K. W. CASSEL
Keyword(s):  
Boundary Layer ◽  
Large Scale ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Recirculation Region ◽  
Small Scale ◽  
Navier Stokes Equations ◽  
Scale Interaction

Numerical solutions of the unsteady Navier–Stokes equations are considered for the flow induced by a thick-core vortex convecting along a surface in a two-dimensional incompressible flow. The presence of the vortex induces an adverse streamwise pressure gradient along the surface that leads to the formation of a secondary recirculation region followed by a narrow eruption of near-wall fluid in solutions of the unsteady boundary-layer equations. The locally thickening boundary layer in the vicinity of the eruption provokes an interaction between the viscous boundary layer and the outer inviscid flow. Numerical solutions of the Navier–Stokes equations show that the interaction occurs on two distinct streamwise length scales depending upon which of three Reynolds-number regimes is being considered. At high Reynolds numbers, the spike leads to a small-scale interaction; at moderate Reynolds numbers, the flow experiences a large-scale interaction followed by the small-scale interaction due to the spike; at low Reynolds numbers, large-scale interaction occurs, but there is no spike or subsequent small-scale interaction. The large-scale interaction is found to play an essential role in determining the overall evolution of unsteady separation in the moderate-Reynolds-number regime; it accelerates the spike formation process and leads to formation of secondary recirculation regions, splitting of the primary recirculation region into multiple corotating eddies and ejections of near-wall vorticity. These eddies later merge prior to being lifted away from the surface and causing detachment of the thick-core vortex.

Essential Ingredients for the Computation of Steady and Unsteady Blade Boundary Layers

1991 ◽  
Vol 113 (4) ◽  
pp. 608-616 ◽  
Author(s):  
H. M. Jang ◽  
J. A. Ekaterinaris ◽  
M. F. Platzer ◽  
T. Cebeci
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Transitional Region ◽  
Low Reynolds Numbers ◽  
Flow Equations ◽  
Low Reynolds

Two methods are described for calculating pressure distributions and boundary layers on blades subjected to low Reynolds numbers and ramp-type motion. The first is based on an interactive scheme in which the inviscid flow is computed by a panel method and the boundary layer flow by an inverse method that makes use of the Hilbert integral to couple the solutions of the inviscid and viscous flow equations. The second method is based on the solution of the compressible Navier–Stokes equations with an embedded grid technique that permits accurate calculation of boundary layer flows. Studies for the Eppler-387 and NACA-0012 airfoils indicate that both methods can be used to calculate the behavior of unsteady blade boundary layers at low Reynolds numbers provided that the location of transition is computed with the en method and the transitional region is modeled properly.

scholarly journals Finite-amplitude steady waves in plane viscous shear flows

1985 ◽  
Vol 160 ◽  
pp. 281-295 ◽  
Author(s):  
F. A. Milinazzo ◽  
P. G. Saffman
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Navier Stokes ◽  
Unidirectional Flow ◽  
Viscous Shear ◽  
Finite Amplitude Waves

Computations of two-dimensional solutions of the Navier–Stokes equations are carried out for finite-amplitude waves on steady unidirectional flow. Several cases are considered. The numerical method employs pseudospectral techniques in the streamwise direction and finite differences on a stretched grid in the transverse direction, with matching to asymptotic solutions when unbounded. Earlier results for Poiseuille flow in a channel are re-obtained, except that attention is drawn to the dependence of the minimum Reynolds number on the physical constraint of constant flux or constant pressure gradient. Attempts to calculate waves in Couette flow by continuation in the velocity of a channel wall fail. The asymptotic suction boundary layer is shown to possess finite-amplitude waves at Reynolds numbers orders of magnitude less than the critical Reynolds number for linear instability. Waves in the Blasius boundary layer and unsteady Rayleigh profile are calculated by employing the artifice of adding a body force to cancel the spatial or temporal growth. The results are verified by comparison with perturbation analysis in the vicinity of the linear-instability critical Reynolds numbers.

Non-uniqueness in wakes and boundary layers

1984 ◽  
Vol 391 (1800) ◽  
pp. 1-26 ◽  
Keyword(s):  
Boundary Layer ◽  
Large Scale ◽  
Reynolds Numbers ◽  
Near Wake ◽  
Algebraic Decay ◽  
Scale Separation ◽  
Boundary Layer Equations ◽  
Classical Boundary ◽  
Streamlined Flow ◽  
High Reynolds

In streamlined flow past a flat plate aligned with a uniform stream, it is shown that ( a ) the Goldstein near-wake and ( b ) the Blasius boundary layer are non-unique solutions locally for the classical boundary layer equations, whereas ( c ) the Rott-Hakkinen very-near-wake appears to be unique. In each of ( a ) and ( b ) an alternative solution exists, which has reversed flow and which apparently cannot be discounted on immediate grounds. So, depending mainly on how the alternatives for ( a ), ( b ) develop downstream, the symmetric flow at high Reynolds numbers could have two, four or more steady forms. Concerning non-streamlined flow, for example past a bluff obstacle, new similarity forms are described for the pressure-free viscous symmetric closure of a predominantly slender long wake beyond a large-scale separation. Features arising include non-uniqueness, singularities and algebraic behaviour, consistent with non-entraining shear layers with algebraic decay. Non-uniqueness also seems possible in reattachment onto a solid surface and for non-symmetric or pressure-controlled flows including the wake of a symmetric cascade.

scholarly journals Assessment of inner–outer interactions in the urban boundary layer using a predictive model

2019 ◽  
Vol 875 ◽  
pp. 44-70 ◽  
Author(s):  
Karin Blackman ◽  
Laurent Perret ◽  
Romain Mathis
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Predictive Model ◽  
Boundary Layers ◽  
Large Scale ◽  
Reynolds Numbers ◽  
Limited Range ◽  
Rough Wall ◽  
Wall Boundary ◽  
Layer Flow

Urban-type rough-wall boundary layers developing over staggered cube arrays with plan area packing density, $\unicode[STIX]{x1D706}_{p}$, of 6.25 %, 25 % or 44.4 % have been studied at two Reynolds numbers within a wind tunnel using hot-wire anemometry (HWA). A fixed HWA probe is used to capture the outer-layer flow while a second moving probe is used to capture the inner-layer flow at 13 wall-normal positions between $1.25h$ and $4h$ where $h$ is the height of the roughness elements. The synchronized two-point HWA measurements are used to extract the near-canopy large-scale signal using spectral linear stochastic estimation and a predictive model is calibrated in each of the six measurement configurations. Analysis of the predictive model coefficients demonstrates that the canopy geometry has a significant influence on both the superposition and amplitude modulation. The universal signal, the signal that exists in the absence of any large-scale influence, is also modified as a result of local canopy geometry suggesting that although the nonlinear interactions within urban-type rough-wall boundary layers can be modelled using the predictive model as proposed by Mathis et al. (J. Fluid Mech., vol. 681, 2011, pp. 537–566), the model must be however calibrated for each type of canopy flow regime. The Reynolds number does not significantly affect any of the model coefficients, at least over the limited range of Reynolds numbers studied here. Finally, the predictive model is validated using a prediction of the near-canopy signal at a higher Reynolds number and a prediction using reference signals measured in different canopy geometries to run the model. Statistics up to the fourth order and spectra are accurately reproduced demonstrating the capability of the predictive model in an urban-type rough-wall boundary layer.

Rayleigh number scaling in numerical convection

1996 ◽  
Vol 310 ◽  
pp. 139-179 ◽  
Author(s):  
Robert M. Kerr
Keyword(s):  
Boundary Layer ◽  
Rayleigh Number ◽  
Large Scale ◽  
Stokes Equations ◽  
Good Evidence ◽  
Navier Stokes ◽  
Fixed Temperature ◽  
Navier Stokes Equations ◽  
Local Boundary ◽  
Velocity Boundary Layer

Using direct simulations of the incompressible Navier-Stokes equations with rigid upper and lower boundaries at fixed temperature and periodic sidewalls, scaling with respect to Rayleigh number is determined. At large aspect ratio (6:6:1) on meshes up to 288 × 288 × 96, a single scaling regime consistent with the properties of ‘hard’ convective turbulence is found for Pr = 0.7 between Ra = 5 × 104 and Ra = 2 × 107. The properties of this regime include Nu ∼ RaβT with βT = 0.28 ≈ 2/7, exponential temperature distributions in the centre of the cell, and velocity and temperature scales consistent with experimental measurements. Two velocity boundary-layer thicknesses are identified, one outside the thermal boundary layer that scales as Ra−1/7 and the other within it that scales as Ra−3/7. Large-scale shears are not observed; instead, strong local boundary-layer shears are observed in regions between incoming plumes and an outgoing network of buoyant sheets. At the highest Rayleigh number, there is a decade where the energy spectra are close to k−5/3 and temperature variance spectra are noticeably less steep. It is argued that taken together this is good evidence for ‘hard’ turbulence, even if individually each of these properties might have alternative explanations.

Identification of Bypass Transition Onset Markers Using Direct Numerical Simulation

2018 ◽  
Vol 140 (11) ◽  
Author(s):  
Shanti Bhushan ◽  
D. Keith Walters ◽  
S. Muthu ◽  
Crystal L. Pasiliao
Keyword(s):  
Numerical Simulation ◽  
Boundary Layer ◽  
Direct Numerical Simulation ◽  
Large Scale ◽  
General Purpose ◽  
Navier Stokes ◽  
Bypass Transition ◽  
Flow Parameters ◽  
Rans Models ◽  
Large Scale Flow

Efficacy of several large-scale flow parameters as transition onset markers are evaluated using direct numerical simulation (DNS) of boundary layer bypass transition. Preliminary results identify parameters (k2D/ν) and u′/U∞ to be a potentially reliable transition onset marker, and their critical values show less than 15% variation in the range of Re and turbulence intensity (TI). These parameters can be implemented into general-purpose physics-based Reynolds-averaged Navier–Stokes (RANS) models for engineering applications.

Potential Flow Field Generated by Downstream Moving Bars and Propagated on a Turbine Blade at Low Reynolds Number

2011 ◽  
Author(s):  
Ve´ronique Penin ◽  
Pascale Kulisa ◽  
Franc¸ois Bario
Keyword(s):  
Boundary Layer ◽  
Large Scale ◽  
Numerical Study ◽  
Three Dimensional ◽  
Potential Effect ◽  
Navier Stokes ◽  
Turbine Cascade ◽  
Wake Interaction ◽  
Rotor Stator Interaction ◽  
The Stability

During the last few decades, the size and weight of turbo-machinery have been continuously reduced. However, by decreasing the distance between rows, rotor-stator interaction is strengthened. Two interactions now have the same magnitude: wake interaction and potential effect. Studying this effect is essential to understand rotor-stator interactions. Indeed, this phenomenon influences the whole flow, including the boundary layer of the upstream and downstream blades, ergo the stability of the flow and the efficiency of the machine. A large scale turbine cascade followed by a specially designed rotating cylinder system is used. Synchronised velocity LDA measurements on the vane profile show the flow and boundary layer behavior due to the moving bars. To help the general understanding and to corroborate our experimental results, numerical investigations are carried out with an unsteady three dimensional Navier-Stokes code. Moreover, the numerical study informs about the potential disturbance to the whole flow of the cascade.

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