Unsteady Viscous Flow on Oscillating Airfoils

Chapter 10 - Companion Computer Programs

In this paper, the effects of adiabatic and isothermal conditions on the statistics in compressible turbulent channel flow are investigated using direct numerical simulation (DNS). DNS of two compressible turbulent channel flows (Cases 1 and 2) are performed using a mixed Fourier Galerkin and B-spline collocation method. Case 1 is compressible turbulent channel flow between isothermal walls, which corresponds to DNS performed by Coleman et al. (1995). Case 2 is the flow between adiabatic and isothermal walls. The flow of Case 2 can be a very useful framework for the present objective, since it is the simplest turbulent channel flow with an adiabatic wall and provides ideal information for modelling the compressible turbulent flow near the adiabatic wall. Note that compressible turbulent channel flow between adiabatic walls is not stationary if there is no sink of heat. In Cases 1 and 2, the Mach number based on the bulk velocity and sound speed at the isothermal wall is 1.5, and the Reynolds number based on the bulk density, bulk velocity, channel half-width, and viscosity at the isothermal wall is 3000. To compare compressible and incompressible turbulent flows, DNS of two incompressible turbulent channel flows with passive scalar transport (Cases A and B) are performed using a mixed Fourier Galerkin and Chebyshev tau method. The wall boundary conditions of Cases A and B correspond to those of Cases 1 and 2, respectively. Case A corresponds to the DNS of Kim & Moin (1989). In Cases A and B, the Reynolds number based on the friction velocity, the channel half-width, and the kinematic viscosity is 150. The mean velocity and temperature near adiabatic and isothermal walls for compressible turbulent channel flow can be explained using the non-dimensional heat flux and the friction Mach number. It is found that Morkovin's hypothesis is not applicable to the near-wall asymptotic behaviour of the wall-normal turbulence intensity even if the variable property effect is taken into account. The mechanism of the energy transfers among the internal energy, mean and turbulent kinetic energiesis investigated, and the difference between the energy transfers near isothermal and adiabatic walls is revealed. Morkovin's hypothesis is not applicable to the correlation coeffcient between velocity and temperature fluctuations near the adiabatic wall.

This paper presents results from a series of numerical studies of various geometry riblets in both laminar and turbulent external flows, which complement results already reported by colleagues at IMST, ONERA-CERT, and UMIST, for V and L-groove riblets in laminar internal and external flows. Although these earlier studies suggested that a 2–3% reduction could be achieved in the latter case, questions remained as to the sensitivity of this finding to further grid refinement, particularly as no drag reduction was indicated for fully developed internal flows. New, higher resolution, CRAY 2 V-groove riblet computations, employing both orthogonal cartesian and conformal grids, have tended to confirm an optimum 2.5% drag reduction for riblets scaling on the boundary layer thickness (rather than wall units as for turbulent flow), and computed mean velocity profiles agreed well with LDA experimental data. Conformal mesh computations for semi-circular U, L, and intermediate form, riblets indicated a similar drag reduction in laminar flow, but also revealed important differences in the spanwise distribution of Cf across these riblets compared to the V-groove type. Subsequent turbulent computations for V and L-groove riblets, employing a conformally transformed mixing length, resulted in mean velocity profiles which again compared well with the limited available turbulent flow data. Parametric computations revealed that the maximum drag reduction was considerably overpredicted for the V-groove riblets, but the optimum performance of the L-groove riblets and the variation in drag reduction with h+ were rather better predicted. Any discrepancies could be associated with secondary motions set-up inside the riblets, since these may be responsible for additional momentum transport, and hence higher Cf, but were not accounted for in the modelling.

Two distinct regimes of tone generation have been isolated for a cavity mounted in a laminar boundary layer in air. The first is a cavity resonance phenomenon similar to that excited by turbulent flow (F. C. De Metz and T. M. Farabee, Paper No. 77-1293, AIAA 4th Aeroacoustics Conference, 1977; S. A. Elder, J. Acoust. Soc. Am. 64, 877 (1978)]. The second is a species of edgetone oscillation which occurs when sufficient damping material is inserted into the cavity to prevent normal mode excitation. By extension of the methods used in earlier studies of turbulent air flow tones, it is possible to model the feedback coupling mechanism for each regime. For the edgetone case, Strouhal number is found to occur close to the value corresponding to maximum instability of the separated shear layer. In the case of resonant cavity oscillation. Strouhal number is somewhat dependent on aspect ratio of the cavity mouth opening. For narrow slot profiles, the resonant oscillation condition is dominated by ξ1 interface waves, resulting in a simple Strouhal number formula. [Work partially supported by a GHR contract with the Naval Ship Systems Command, administered by the David W. Taylor Naval Ship Research and Development Center, Carderock, Maryland.]

The objective is to evaluate the applicability of the penalty method for practically high-Re flows and to investigate the influence of wall model on the quality of the predicted results. This penalty method is introduced at first time on staggered Cartesian grid and then using unstructured mesh. For validation, due to faster convergence, 2D finite volume method in a cartesian grid is used to numerically investigate the incompressible unsteady laminar (Re = 100) and turbulent (Re = 3900) flow around a circular cylinder and the turbulent flow in a channel at \(Re_{u_{\tau}}=395\). 3D simulation is performed on an unstructured grid to simulate the turbulent flow behind a circular cylinder at Re = 3900 without the use of a wall model.

The fast and accurate simulation of laminar and turbulent incompressible flows is crucial in science and engineering. Low-order numerical strategies, particularly finite-volume (FV) methods based on cell-centred and vertex-centred approaches, remain essential in CFD, especially in industrial settings, due to their favourable trade-off between computational cost and accuracy. However, these methods face challenges such as stabilising convective-dominated flows, handling velocity-pressure coupling, and managing numerical flux reconstruction on distorted and stretched grids. The face-centred finite volume (FCFV) method emerges as an alternative to standard FV methods. Derived from a mixed formulation of the discontinuous Galerkin method, FCFV avoids flux reconstruction at cell faces, making its accuracy and convergence nearly insensitive to grid quality. It also satisfies the Ladyzhenskaya-Babuˇska- Brezzi (LBB) condition without special treatment for velocity-pressure coupling. This work showcases the FCFV method for simulating laminar and turbulent incompressible flows for the first time. The formulation is based on the Reynolds-averaged Navier-Stokes (RANS) equations coupled with the negative Spalart-Allmaras (SA) model. Three new convective stabilisations inspired by Riemann solvers are proposed, along with monolithic and staggered solution strategies for the RANS-SA system and two relaxation strategies for pseudo-time marching. A new hybrid pressure FCFV formulation is also introduced to improve the FCFV accuracy in laminar flow simulations by enriching the pressure space through relaxed compressibility conditions. Both FCFV and hybrid pressure FCFV achieve first-order con- vergence of velocity, velocity gradient tensor, and pressure, accurately predicting engineering quantities such as drag and lift on structured and unstructured meshes. By avoiding gradient reconstruction, these methods are less sensitive to mesh quality, even on highly distorted grids. Numerical benchmarks for laminar and turbulent, steady and transient cases assess the performance, accuracy, and robustness of the proposed methodologies. Implemented in Fortran 90, these methods lay the foundation for the integration of FCFV techniques within the CFD community. The work concludes with a detailed discussion of the Fortran 90 FCFV code implementation.

A gas-kinetic scheme coupled with SST model for turbulent flows

An up-down counter, multiplier, digital-to-analog (D/A) converter, and high-speed comparators are employed to achieve an automatic gain control (AGC) circuit, which corrects the pair of characteristic roots of the overall system automatically to the imaginary axis of the complex frequency plane. A negative feedback technique with digital hardware is applied on the loop gain control. No lowpass filter is needed to detect the oscillation amplitude. Thus, this technique is suitable for sinusoidal oscillators with a wide oscillation frequency range. Wien-bridge and phase-shift oscillators with an oscillation frequency range from 17 Hz to 1 MHz are tested with the proposed AGC circuit. The total harmonic distortions of the Wien-bridge sinusoidal oscillator with the proposed AGC circuit are verified to be very small. An application to a variable-frequency sinusoidal oscillator is also described. The experimental results demonstrate the static characteristics and dynamic responses of the overall system.

Monitoring of drinking water (DW) biofilm formation under different process conditions was performed using two distinct bioreactors: a Propella™ and a flow cell system. Biofilms were grown on polyvinyl chloride (PVC) and stainless steel (SS) coupons under laminar (Reynolds number: 2000) and turbulent (Reynolds number: 11000) flow. The parameters analysed were the numbers of total and cultivable bacteria. The impact of different process conditions was assessed after the biofilms reached steady-state. The number of total bacteria was mostly higher than those cultivable. Biofilm steady-state was achieved in 3 days in both bioreactors with adhesion surfaces under turbulent flow. Under laminar flow it was only achieved in 6 days. The numbers of total and cultivable bacteria in turbulent flow-generated biofilms were similar in both bioreactors, regardless of the adhesion surface tested. Under laminar flow, the Propella™ bioreactor allowed the formation of steady-state biofilms with a higher number of total and cultivable bacteria than the flow cell system. Comparing the effects of the flow regime on biofilm accumulation, only turbulent flow-generated biofilms formed on the flow cell system had a higher amount of total and cultivable bacteria than those formed under laminar flow. In terms of adhesion surface effects, a higher number of total and cultivable cells were found on PVC surfaces compared to SS when biofilms were formed in the flow cell system. Biofilm formation on PVC and SS was similar in the Propella™ system for both flow regimes.

In this work, a two-dimensional laminar and turbulent Power law fluid flow passing a square cylinder is numerically investigated. In the investigation, a finite volume code based on the SIMPLEC algorithm and non-staggered grid was used. In the discretization of the convective and diffusive terms, the third-order QUICK and the second-order central difference scheme, respectively were used. The turbulent flow was simulated using the model. Extensive numerical results of the drag coefficient, root mean square of the lift coefficient, Strouhal number, stream functions, pressure coefficient, time-averaged velocities and power law viscosity are presented to determine the influence of the Power law index and Reynolds number. Laminar flows discussed in the present work are within the ranges: and covering shear thinning, Newtonian and shear thickening fluids. Turbulent flows within the ranges of Re=13000 and 22000 and were also evaluated numerically and discussed. Key words: Single square cylinder, turbulence modeling, power law fluid, lift and drag coefficients, Strouhal number.

The swivelling motion of a flexible structure immersed in a flow can become self-excited as a result of different fluid-structure interaction mechanisms. The accurate simulation of these mechanisms still constitutes a challenge with respect to mathematical modelling, numerical discretization, solution techniques, and implementation as software tools on modern computer architectures. Thus, to support the development of numerical codes for fluid structure interaction computations, in the present work an experimental investigation on the two-dimensional self-excited periodic swivelling motion of flexible structures in both laminar and turbulent uniform flows was performed. The investigated structural model consisted of a stainless-steel flexible sheet attached to a cylindrical front body. At the trailing edge of the flexible sheet, a rectangular mass was considered. The entire structure model was free to rotate around an axle located in the central point of the front body. During the experimental investigation, the general character of the elastic-dynamic response of the structure model was studied first. The tests in laminar flows were performed in a polyglycol syrup (dynamic viscosity: 1.64 ×10−4 m{ 2}/s) for a Reynolds number smaller than 270, whereas the tests in turbulent flows were conducted in water for Reynolds numbers up to 44000. In both cases, the maximum incoming velocity tested was about 2 m/s. Subsequently, three specific test cases were selected and characterized in more detail as far as the flow velocity field and structure mechanical behavior are concerned. Thus, the present contribution presents the detailed results obtained at 1.07 m/s and at 1.45 m/s in laminar and at 0.68 m/s in turbulent flows. It also compares the experimental data with numerical results obtained for the same conditions using different simulating approaches. They revealed very good agreement in some of the fluid-structure interaction modes whereas in others deficiencies were observed that need to be analyzed in more detail.

The main objective of this study is to clarify the effect of thermal wall boundary conditions on turbulence statistics and structures in a compressible turbulent flow. This work is an extension of Morinishi et al. (J. Fluid Mech. vol. 502, 2004, p. 273), who performed DNS of compressible turbulent channel flow between adiabatic and isothermal walls at Mach number $M\,{=}\,1.5$ (Case 2). We address the question of whether the modification of turbulence statistics is attributable to the effect of the adiabatic wall boundary condition or the effect of the increase of wall temperature caused by the adiabatic wall boundary condition. New DNS of the compressible turbulent channel flow between isothermal walls with the wall temperature difference at the Mach number $M=1.5$ (Case 1) and DNS of the corresponding incompressible turbulent flow with passive scalar transport (Case I) are performed. The present study shows that the mean temperature profile near the high-temperature wall for Case 1 has an additional maximum due to the friction work, while such an additional maximum does not appear for Cases 2 and I. The additional maximum leads to a corresponding near-wall maximum of temperature fluctuations. We find the direction of energy transfer due to pressure work near the adiabatic wall for Case 2 being opposite to that near the isothermal wall to be due to the effect of the high-temperature wall, not to the effect of the adiabatic wall. These findings are explained by using the budgets of internal energy and temperature variance transport equations.

We give a brief introduction to research on adaptive computational methods for laminar compressible and incompressible flows and then focus on computability and adaptivity for turbulent incompressible flow, where we present a framework for adaptive finite element methods with duality‐baseda posteriorierror control for chosen output quantities of interest. We show in concrete examples that outputs such as mean values in time of drag and lift of a bluff body in a turbulent flow are computable to a tolerance of a few percent, for a simple geometry using some hundred thousand mesh points and for complex geometries using some million mesh points.

Free flow above permeable domains is a common problem in both natural and industrial environments. The classical problem of turbulent flows above forests and urban areas, known as the canopy flow problem (Finnigan 2000), was addressed by hundreds of publications. A library search for the keywords ‘canopy and flow’ in the ISI database returned 1312 results in April 2007 and 1845 results in April 2009, about 40% increase in the last 2years alone. In fact, this search underestimates the actual number of publications since research of flow above permeable domains covers a much wider range of scenarios. Vegetated flows in rivers and wetlands, flow above gravel streambeds, and flow above coral reefs are a few environmental examples; industrial examples include flows in coating processes, chemical reactors, and electronic cooling; and examples of a much smaller length scale include flows above bio-films and inside bio-tissues. Studies of flows over permeable domains include both laminar and turbulent flow regimes. Where most of the porous media studies focus on laminar and creeping flows, terrestrial, ocean, and atmospheric flows are all turbulent. Combinations such as turbulent channel flows above laminar flow inside dense gravel beds are also common. Most of the past and present research activities are often classified by the following four disciplines: porous media flows (e.g., Whitaker 1999), atmospheric canopy flows (e.g., Finnigan 2000), vegetated flow (e.g., Nepf 1999), and fluvial flows. Studies of fluvial flows were recently addressed in a special issue on double averaging applications (Nikora and Rowinski 2008). The following special issue with its 13 publications covers both laminar and turbulent flow regimes while emphasizing on fundamental questions of formulating the transport conditions at the interface between the fluid and the porous domains. The most famous publication on laminar flows above porous interfaces is the study by Beavers and Joseph (1967) (cited 771 times, ISI, April 2009). By investigating the flow above porous domains, they developed a linear relationship between the velocity gradient above the interface and the interface slip velocity. However, numerous studies showed that the Beavers and Joseph condition is not general enough and a coupled solution of both the free flow and

Two solution methods for compressible flows and their application in Large-Eddy Simulation (LES) of turbulent flows are discussed. After a summary of the basic equations and subgrid scale models, the applied discretization methods, solution schemes and boundary conditions with their implementation on high-performance computers are presented. The two discretization methods for the advective terms used in the simulations are a second-order mixed central-upwind discretization based on a AUSM formulation and a higher-order compact finite difference scheme. First, results of the second-order scheme are shown for the case of a turbulent channel and pipe flow, which are only weakly influenced by subgrid scale models, but are in good agreement with DNS and experimental data. Higher-order compact schemes are then used for both laminar and turbulent flows and the results are compared to those from simulations with the second-order scheme. It is found that for the laminar flow and the LES of plane turbulent jets the higher-order compact scheme is not of principal advantage. This is primarily related to the sensitivity of the compact scheme to stretched and curvilinear grids, which requires explicit filtering steps to avoid unphysical oscillations in the solution. Further results are then presented, which are obtained with the computationally cheaper second-order scheme. The results include turbulent flows in pipe bends with different Reynolds and Dean numbers. The solutions show swirl switching with a distinct low frequency peak. Finally, turbulent flows in non-trivial geometries are considered. This includes a first step towards the LES of the flow in an internal combustion engine with four valves and a realistic geometry.