A Functional Relationship for Modeling Laminar to Turbulent Flow Transitions

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
George Papadopoulos
Keyword(s):  
Turbulent Flow ◽  
Functional Relationship ◽  
Stokes Equations ◽  
Initial Conditions ◽  
Flow Region ◽  
Flow Regimes ◽  
Transitional Flow ◽  
Turbulent Regime ◽  
Flow Profile ◽  
Mean Flow

A dimensional analysis which is based on the scaling of the two-dimensional Navier–Stokes equations is presented for correlating bulk flow characteristics arising from a variety of initial conditions. The analysis yields a functional relationship between the characteristic variable of the flow region and the Reynolds number for each of the two independent flow regimes, laminar and turbulent. A linear relationship is realized for the laminar regime, while a nonlinear relationship is realized for the turbulent regime. Both relationships incorporate mass-flow profile characteristics to capture the effects of initial conditions (mean flow and turbulence) on the variation of the characteristic variable. The union of these two independent relationships is formed leveraging the concept of flow intermittency to yield a generic functional relationship that incorporates transitional flow effects and fully encompasses solutions spanning the laminar to turbulent flow regimes. Empirical models to several common flows are formed to demonstrate the engineering potential of the proposed functional relationship.

A Functional Scaling Relationship for Laminar to Turbulent Flow Regimes

2011 ◽  
Author(s):  
George Papadopoulos
Keyword(s):  
Turbulent Flow ◽  
Stokes Equations ◽  
Initial Conditions ◽  
Flow Characteristics ◽  
Flow Region ◽  
Flow Regimes ◽  
Transitional Flow ◽  
Turbulent Regime ◽  
Flow Profile ◽  
Scaling Relationship

A dimensional analysis that is based on the scaling of the two-dimensional Navier-Stokes equations is presented for correlating bulk flow characteristics arising from a variety of initial conditions. The analysis yields a functional relationship between the characteristic variable of the flow region and the Reynolds number for each of the two independent flow regimes. A linear relationship is realized for the laminar regime, while a nonlinear relationship is realized for the turbulent regime. Both relationships incorporate mass-flow profile characteristics to fully capture the effects of initial conditions on the variation of the characteristic variables. The union of these two independent relationships is formed utilizing the concept of flow intermittency to further expand into a generic scaling relationship that incorporates transitional flow effects to fully encompass solutions spanning the laminar to turbulent flow regimes. The results of the analysis are discussed within the context of several flow phenomena (e.g. pipe flow, jet flow & separated flow) resulting from various initial and boundary conditions.

Characteristics of the leading Lyapunov vector in a turbulent channel flow

2018 ◽  
Vol 849 ◽  
pp. 942-967 ◽  
Author(s):  
Nikolay Nikitin
Keyword(s):  
Turbulent Flow ◽  
Buffer Layer ◽  
Stokes Equations ◽  
Initial Conditions ◽  
Spatial Scales ◽  
Plane Channel ◽  
Base Flow ◽  
Lyapunov Vector ◽  
Perturbation Growth ◽  
Near Wall

The values of the highest Lyapunov exponent (HLE)$\unicode[STIX]{x1D706}_{1}$for turbulent flow in a plane channel at Reynolds numbers up to$Re_{\unicode[STIX]{x1D70F}}=586$are determined. The instantaneous and statistical properties of the corresponding leading Lyapunov vector (LLV) are investigated. The LLV is calculated by numerical solution of the Navier–Stokes equations linearized about the non-stationary base solution corresponding to the developed turbulent flow. The base turbulent flow is calculated in parallel with the calculation of the evolution of the perturbations. For arbitrary initial conditions, the regime of exponential growth${\sim}\exp (\unicode[STIX]{x1D706}_{1}t)$which corresponds to the approaching of the perturbation to the LLV is achieved already at$t^{+}<50$. It is found that the HLE increases with increasing Reynolds number from$\unicode[STIX]{x1D706}_{1}^{+}\approx 0.021$at$Re_{\unicode[STIX]{x1D70F}}=180$to$\unicode[STIX]{x1D706}_{1}^{+}\approx 0.026$at$Re_{\unicode[STIX]{x1D70F}}=586$. The LLV structures are concentrated mainly in a region of the buffer layer and are manifested in the form of spots of increased fluctuation intensity localized both in time and space. The root-mean-square (r.m.s.) profiles of the velocity and vorticity intensities in the LLV are qualitatively close to the corresponding profiles in the base flow with artificially removed near-wall streaks. The difference is the larger concentration of LLV perturbations in the vicinity of the buffer layer and a relatively larger (by approximately 80 %) amplitude of the vorticity pulsations. Based on the energy spectra of velocity and vorticity pulsations, the integral spatial scales of the LLV structures are determined. It is found that LLV structures are on average twice narrower and twice shorter than the corresponding structures of the base flow. The contribution of each of the terms entering into the expression for the production of the perturbation kinetic energy is determined. It is shown that the process of perturbation development is essentially dictated by the inhomogeneity of the base flow, as well as by the presence of transversal motion in it. Neglecting of these factors leads to a significant underestimation of the perturbation growth rate. The presence of near-wall streaks in the base flow, on the contrary, does not play a significant role in the development of the LLV perturbations. Artificial removal of streaks from the base flow does not change the character of the perturbation growth.

Vorticity Transport Analysis of Turbulent Flows

1995 ◽  
Vol 117 (3) ◽  
pp. 410-416 ◽  
Author(s):  
J. J. Gorski ◽  
P. S. Bernard
Keyword(s):  
Turbulent Flows ◽  
Transport Theory ◽  
Stokes Equations ◽  
Transport Model ◽  
Plate Boundary ◽  
Separated Flow ◽  
Flow Region ◽  
Leeward Side ◽  
Mean Flow ◽  
Vorticity Transport

Turbulence closure for the Reynolds averaged Navier-Stokes equations based on vorticity transport theory is investigated. General expressions for the vorticity transport correlation terms in arbitrary two-dimensional mean flows are derived. Direct numerical simulation data for flow in a channel is used to evaluate the modeled terms and set unknown scales. Results are presented for a channel, flat plate boundary layer, and flow over a hill. The computed mean flow and kinetic energy compares well with numerical and physical experiments. The vorticity transport model appears to perform better than conventional Boussinesq eddy viscosity Reynolds stress models near the separated flow region on the leeward side of the hill.

Compressibility effects in Rayleigh–Taylor instability-induced flows

2010 ◽  
Vol 368 (1916) ◽  
pp. 1681-1704 ◽  
Author(s):  
S. Gauthier ◽  
B. Le Creurer
Keyword(s):  
Compressible Flows ◽  
Stokes Equations ◽  
Initial Conditions ◽  
Density Fluctuations ◽  
Taylor Instability ◽  
Turbulent Regime ◽  
Developed Turbulence ◽  
Perfect Fluids ◽  
Rayleigh Taylor Instability ◽  
Compressibility Effects

We present a tentative review of compressibility effects in Rayleigh–Taylor instability-induced flows. The linear, nonlinear and turbulent regimes are considered. We first make the classical distinction between the static compressibility or stratification, and the dynamic compressibility owing to the finite speed of sound. We then discuss the quasi-incompressible limits of the Navier–Stokes equations (i.e. the low-Mach number, anelastic and Boussinesq approximations). We also review some results about stratified compressible flows for which instability criteria have been derived rigorously. Two types of modes, convective and acoustic, are possible in these flows. Linear stability results for perfect fluids obtained from an analytical approach, as well as viscous fluid results obtained from numerical approaches, are also reviewed. In the turbulent regime, we introduce Chandrasekhar’s observation that the largest structures in the density fluctuations are determined by the initial conditions. The effects of compressibility obtained by numerical simulations in both the nonlinear and turbulent regimes are discussed. The modifications made to statistical models of fully developed turbulence in order to account for compressibility effects are also treated briefly. We also point out the analogy with turbulent compressible Kelvin–Helmholtz mixing layers and we suggest some lines for further investigations.

Navier-Stokes Computations of the NREL Airfoil Using a κ — ω Turbulent Model at High Angles of Attack

1995 ◽  
Vol 117 (4) ◽  
pp. 304-310 ◽  
Author(s):  
S. L. Yang ◽  
Y. L. Chang ◽  
O. Arici
Keyword(s):  
Turbulent Flow ◽  
Stokes Equations ◽  
Solid Wall ◽  
Wind Tunnel Test ◽  
Navier Stokes ◽  
Mean Flow ◽  
Approximate Factorization ◽  
Navier Stokes Equations ◽  
Reynolds Number Effects ◽  
Model Equations

This paper presents a two-dimensional numerical simulation of the turbulent flow fields for the NREL (National Renewable Energy Laboratory) S809 airfoil. The flow is modeled as steady, viscous, turbulent, and incompressible. The pseudo-compressible formulation is used for the time-averaged Navier-Stokes equations so that a time marching scheme developed for the compressible flow can be applied directly. The turbulent flow is simulated using Wilcox’s modified κ — ω model to account for the low Reynolds number effects near a solid wall and the model’s sensitivity to the freestream conditions. The governing equations are solved by an implicit approximate-factorization scheme. To correctly model the convection terms in the mean-flow and turbulence model equations, the symmetric TVD (Total Variational Diminishing) scheme is incorporated. The methodology developed is then applied to analyze the NREL S809 airfoil at various angles of attack (α) from 1 to 45 degrees. The accuracy of the numerical results is compared with the available Delft wind tunnel test data. For comparison, two Eppler code results at low angles of attack are also included. Depending on the value of α, preliminary results show excellent to fairly good agreement with the experimental data. Directions for future work are also discussed.

A resonant wave theory

1999 ◽  
Vol 395 ◽  
pp. 237-251 ◽  
Author(s):  
LUN-SHIN YAO
Keyword(s):  
Stokes Equations ◽  
Flow Instability ◽  
Initial Conditions ◽  
Laminar Flows ◽  
Navier Stokes ◽  
Real Interval ◽  
Mean Flow ◽  
Navier Stokes Equations ◽  
The Mean ◽  
The Waves

Analysis is used to show that a solution of the Navier–Stokes equations can be computed in terms of wave-like series, which are referred to as waves below. The mean flow is a wave of infinitely long wavelength and period; laminar flows contain only one wave, i.e. the mean flow. With a supercritical instability, there are a mean flow, a dominant wave and its harmonics. Under this scenario, the amplitude of the waves is determined by linear and nonlinear terms. The linear case is the target of flow-instability studies. The nonlinear case involves energy transfer among the waves satisfying resonance conditions so that the wavenumbers are discrete, form a denumerable set, and are homeomorphic to Cantor's set of rational numbers. Since an infinite number of these sets can exist over a finite real interval, nonlinear Navier–Stokes equations have multiple solutions and the initial conditions determine which particular set will be excited. Consequently, the influence of initial conditions can persist forever. This phenomenon has been observed for Couette–Taylor instability, turbulent mixing layers, wakes, jets, pipe flows, etc. This is a commonly known property of chaos.

scholarly journals Unified Friction Formulation from Laminar to Fully Rough Turbulent Flow

2018 ◽  
Vol 8 (11) ◽  
pp. 2036 ◽  
Author(s):  
Dejan Brkić ◽  
Pavel Praks
Keyword(s):  
Laminar Flow ◽  
Turbulent Flow ◽  
Pipe Flow ◽  
Flow Regimes ◽  
Hydraulic Model ◽  
Friction Model ◽  
Newtonian Fluids ◽  
Turbulent Regime ◽  
Flow Friction ◽  
Switching Functions

This paper provides a new unified formula for Newtonian fluids valid for all pipe flow regimes from laminar to fully rough turbulent flow. This includes laminar flow; the unstable sharp jump from laminar to turbulent flow; and all types of turbulent regimes, including the smooth turbulent regime, the partial non-fully developed turbulent regime, and the fully developed rough turbulent regime. The new unified formula follows the inflectional form of curves suggested in Nikuradse’s experiment rather than the monotonic shape proposed by Colebrook and White. The composition of the proposed unified formula uses switching functions and interchangeable formulas for the laminar, smooth turbulent, and fully rough turbulent flow regimes. Thus, the formulation presented below represents a coherent hydraulic model suitable for engineering use. This new flow friction model is more flexible than existing literature models and provides smooth and computationally cheap transitions between hydraulic regimes.

scholarly journals Unified Friction Formulation from Laminar to Fully Rough Turbulent Flow

2018 ◽  
Author(s):  
Dejan Brkić ◽  
Pavel Praks
Keyword(s):  
Turbulent Flow ◽  
Pipe Flow ◽  
Flow Regimes ◽  
Hydraulic Model ◽  
Newtonian Fluids ◽  
Turbulent Regime ◽  
Switching Functions

This paper gives a new unified formula for the Newtonian fluids valid for all pipe flow regimes from laminar to the fully rough turbulent. It includes laminar, unstable sharp jump from laminar to turbulent, and all types of the turbulent regimes: smooth turbulent regime, partial non-fully developed turbulent and fully developed rough turbulent regime. The formula follows the inflectional form of curves as suggested in Nikuradse&rsquo;s experiment rather than monotonic shape proposed by Colebrook and White. The composition of the proposed unified formula consists of switching functions and of the interchangeable formulas for laminar, smooth turbulent and fully rough turbulent flow. The proposed switching functions provide a smooth and a computationally cheap transition among hydraulic regimes. Thus, the here presented formulation represents a coherent hydraulic model suitable for engineering use. The model is compared to existing literature models, and shows smooth and computationally cheap transitions among hydraulic regimes.

scholarly journals Experiments on low-Reynolds-number turbulent flow through a square duct

2016 ◽  
Vol 798 ◽  
pp. 398-410 ◽  
Author(s):  
Bayode E. Owolabi ◽  
Robert J. Poole ◽  
David J. C. Dennis
Keyword(s):  
Flow Field ◽  
Turbulent Flow ◽  
Probability Density ◽  
Reynolds Numbers ◽  
Probability Density Functions ◽  
Density Functions ◽  
Turbulent Regime ◽  
Mean Flow ◽  
Square Duct ◽  
Low Reynolds

Previous experimental studies on turbulent square duct flow have focused mainly on high Reynolds numbers for which a turbulence-induced eight-vortex secondary flow pattern exists in the cross-sectional plane. More recently, direct numerical simulations (DNS) have revealed that the flow field at Reynolds numbers close to transition can be very different; the flow in this ‘marginally turbulent’ regime alternating between two states characterised by four vortices. In this study, we experimentally investigate the onset criteria for transition to turbulence in square ducts. In so doing, we highlight the potential importance of Coriolis effects on this process for low-Ekman-number flows. We also present experimental data on the mean flow properties and turbulence statistics in both marginally and fully turbulent flow at relatively low Reynolds numbers using laser Doppler velocimetry. Results for both flow categories show good agreement with DNS. The switching of the flow field between two flow states at marginally turbulent Reynolds numbers is confirmed by bimodal probability density functions of streamwise velocity at certain distances from the wall as well as joint probability density functions of streamwise and wall normal velocities which feature two peaks highlighting the two states.

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