Unsteady Case Research Articles

On the Use of Different Sets of Variables for Solving Unsteady Inviscid Flows with an Implicit Discontinuous Galerkin Method

This article presents a comparison between the performance obtained by using a spatial discretization of the Euler equations based on a high-order discontinuous Galerkin (dG) method and different sets of variables. The sets of variables investigated are as follows: (1) conservative variables; (2) primitive variables based on pressure and temperature; (3) primitive variables based on the logarithms of pressure and temperature. The solution is advanced in time by using a linearly implicit high-order Rosenbrock-type scheme. The results obtained using the different sets are assessed across several canonical unsteady test cases, focusing on the accuracy, conservation properties and robustness of each discretization. In order to cover a wide range of physical flow conditions, the test-cases considered here are (1) the isentropic vortex convection, (2) the Kelvin–Helmholtz instability and (3) the Richtmyer–Meshkov instability.

Analogical elucidation of dusty‐hybrid nanofluid flow through the microchannel: An unsteady case

AbstractThis work is the exploration of unsteady flow of dusty fluid in the horizontal microchannel when the channel plates are susceptible to linear radiation. The flow of the dusty fluid is in the existence of magnetic field. Along with dusty fluid, there are immersions of two different types of nanoparticles namely ferrous oxide and silver . The study gives the comparison of nanofluid phase with dusty fluid phase; hybrid nanofluid phase with dusty fluid phase separately. The modeled partial differential equations are non‐dimensionalized and pdsolve command of Maple software tackles these formed partial differential equations directly. Comparative study of nanofluid‐dusty fluid and hybrid nanofluid‐dusty fluid for the flow in microchannel is to elucidate the fact of heat transfer enhancement upon appropriate insertion of nanoparticles in the base fluid; especially the nanoparticles possessing paramagnetic effect for the better performance on the application of the magnetic field. Results deduce that the fluid flow and heat transmission are known to progress with time. Also, the fluid phase is known to reach the steady state earlier than the dust phase. The nanoparticle volume fraction is known to deplete the temperature in both the hybrid fluid phase and nanofluid phase. The particle concentration parameter diminishes the velocity profile, both for fluid phase and dust phase.

Stabilized weighted reduced order methods for parametrized advection-dominated optimal control problems governed by partial differential equations with random inputs

Abstract In this work, we analyze Parametrized Advection-Dominated distributed Optimal Control Problems with random inputs in a Reduced Order Model (ROM) context. All the simulations are initially based on a finite element method (FEM) discretization; moreover, a space-time approach is considered when dealing with unsteady cases. To overcome numerical instabilities that can occur in the optimality system for high values of the Péclet number, we consider a Streamline Upwind Petrov–Galerkin technique applied in an optimize-then-discretize approach. We combine this method with the ROM framework in order to consider two possibilities of stabilization: Offline-Only stabilization and Offline-Online stabilization. Moreover we consider random parameters and we use a weighted Proper Orthogonal Decomposition algorithm in a partitioned approach to deal with the issue of uncertainty quantification. Several quadrature techniques are used to derive weighted ROMs: tensor rules, isotropic sparse grids, Monte-Carlo and quasi Monte-Carlo methods. We compare all the approaches analyzing relative errors between the FEM and ROM solutions and the computational efficiency based on the speedup-index.

Wind tunnel wall interference correction for transonic airfoils with data-reduced ensemble Kalman filter

The uncertain factors in wind tunnel experiments, especially the interference effects, will cause the flow around the test model to differ from that under the free flow condition. The wind tunnel wall interference effects are challenging to be eliminated, especially in transonic flow regions where classical linear correction methods are ineffective. To address this issue, an efficient wind tunnel wall interference correction framework based on data assimilation is proposed for steady and shock buffet flows around the airfoils in the transonic region in this work. The ensemble Kalman filter (EnKF) is used to find the combination of inflow conditions, which minimizes the differences between the pressure coefficients measured in the wind tunnel experiment and those estimated by computational fluid dynamics (CFD). The polynomial chaos expansion method is used to propagate uncertainty in the forecast step of the EnKF in order to improve efficiency. Typical wind tunnel wall interference correction problems are studied, including two steady cases: the transonic flow around the Royal Aircraft Establishment 2822 (RAE2822) and the National Aerospace Laboratory 7301 (NLR7301) airfoils, and one unsteady case: the transonic shock buffet on the National Aeronautics and Space Administration supercritical (phase 2)-0714 (NASA SC(2)-0714) airfoil. It is demonstrated that with the corrected Mach number and angle of attack, the estimated CFD results are in better agreement with the measured experimental data than traditional linear methods and the computational efficiency is improved compared to the original EnKF.

Model to identify comfortable cloth fabric for human body during thermal stress due to physical exercise

Physical exercise enhances heat generation in human skin and subcutaneous tissues causing thermal stress. This thermal stress causes discomfort in the human body and deteriorates the physical performance of the human subjects. Appropriate choice of cloth for outer covering of human body is necessary to prevent this to provide comfort during the thermal stress. A mathematical model is proposed in this study to analyze the thermal stresses in human skin and subcutaneous tissues covered with various different cloth fabrics. The model considers four compartments namely cloth, epidermis, dermis, subcutaneous tissues. The bioheat equation along with appropriate boundary condition is employed in this model for one dimensional unsteady state case. The numerical simulation is performed using forward and central finite difference formulas. The findings have been utilized to examine how different types of fabric impact thermal stress experienced by the layers of human skin during physical activity. The polyester is found to be superior in comparison to cotton fabric as it shows less thermal stress and thus can be concluded to be more comfortable as a dress material.

Velocity-field based load estimation of an unsteady actuator disc

Traditional methods for determining the load of an object in unsteady cases, such as Control-volume Momentum Integration (CMI) and Noca methods, encounter challenges related to near-body acceleration and pressure estimation. This prompts the need for innovative techniques to overcome these limitations. This study introduces a novel vorticity-based approach to expand classical load determination methods. The numerical results of an unsteady actuator disc (AD) CFD simulation are used to verify and quantify the uncertainty of the three methods. In the AD model, the load is prescribed and used as a source term. The velocity and pressure fields are solved from the Navier-Stokes equations using the open-source CFD package OpenFOAM, which offers a robust validation framework. Systematic examination of the robustness and accuracy of the three methods across various load and motion unsteady scenarios reveals that the proposed vorticity-based method excels in steady-state scenarios, exhibiting the highest level of robustness and accuracy. In contrast, the traditional CMI method proves more robust and accurate in pure motion cases. In scenarios involving static and moving AD with load variations, the new vorticity-based method demonstrates superior robustness and accuracy, mainly when moderate vorticity dynamics. Conversely, in cases with increased vorticity dynamics in the wake, the accuracy of the vorticity-based method decreases, with the CMI method showing superiority.

Spherical partial differential equation with non‐constant coefficients for modeling of nonlinear unsteady heat conduction in functionally graded materials

AbstractThe objective of this research paper is to propose an exact solution for resolving the transient conduction in a 2D sphere. The coefficients of governing equation are varied according to the material properties. The thermo‐physical properties are regarded as functions that follow a power‐law relationship concerning the radial direction. Both radial and angular thermal conductivity coefficients change with radius. Thermal boundary conditions are considered in a general state, which can cover different thermal conditions, including Dirichlet, Neumann, and Convection surface conditions. Laplace transform and Meromorphic function methods are used in the solution approach to the current unsteady problem. Two unsteady case studies with complex boundary conditions have been considered to show the credibility of the current solution. The results of both case studies have been successfully validated. The results confirm the high capability of the present solution in solving unsteady thermal problems of functionally graded materials in spherical coordinates.

A one-dimensional computational model for blood flow in an elastic blood vessel with a rigid catheter.

Strokes are one of the leading causes of death in the United States. Stroke treatment involves removal or dissolution of the obstruction (usually a clot) in the blocked artery by catheter insertion. A computer simulation to systematically plan such patient-specific treatments needs a network of about 105 blood vessels including collaterals. The existing computational fluid dynamic (CFD) solvers are not employed for stroke treatment planning as they are incapable of providing solutions for such big arterial trees in a reasonable amount of time. This work presents a novel one-dimensional mathematical formulation for blood flow modeling in an elastic blood vessel with a centrally placed rigid catheter. The governing equations are first-order hyperbolic partial differential equations, and the hypergeometric function needs to be computed to obtain the characteristic system of these hyperbolic equations. We employed the Discontinuous Galerkin method to solve the hyperbolic system and validated the implementation by comparing it against a well-established 3D CFD solver using idealized vessels and a realistic truncated arterial network. The results showed clinically insignificant differences in steady flow cases, with overall variations between 1D and 3D models remaining below 10%. Additionally, the solver accurately captured wave reflection phenomena at domain discontinuities in unsteady cases. A primary advantage of this model over 3D solvers is its ease in obtaining a discretized geometry of complex vasculatures with multiple arterial branches. Thus, the 1D computational model offers good accuracy and applicability in simulating complex vasculatures, demonstrating promising potential for investigating patient-specific endovascular interventions in strokes.

Unsteady three-dimensional rotational flamelets

A new unsteady flamelet model is developed to be used for sub-grid modeling and coupling with a resolved flow description for turbulent combustion. Difficulties with prior unsteady flamelet models are identified. The model extends the quasi-steady rotational flamelet model, which differs from prior models in several critical ways: (i) the effects of shear strain and vorticity are determined, in addition to normal-strain-rate impacts; (ii) the strain rates and vorticity are determined from the conditions of the environment surrounding the flamelet without a contrived progress variable; (iii) the flamelet model is physically three-dimensional but reduced to a one-dimensional, unsteady formulation using similarity; (iv) variable density is fully addressed in the flamelet model; and (v) non-premixed flames, premixed flames, or multi-branched flame structures are determined rather than prescribed. For both quasi-steady and unsteady cases, vorticity creates a centrifugal force on the flamelet counterflow that modifies the transport rates and burning rate. In the unsteady scenario, new unsteady boundary conditions must be formulated to be consistent with the unsteady equations for the rotating counterflow. Eight boundary values on inflowing scalar and velocity properties and vorticity will satisfy four specific relations and, therefore, cannot all be arbitrarily specified. The temporal variation of vorticity is connected to the variation of applied normal strain rate through the conservation principle for angular momentum. Limitations on the model concerning fluctuation of the interfacial plane are identified and conditions under which interfacial plane fluctuation is negligible are explained. An example of a rotating flamelet counterflow with oscillatory behavior is examined with linearization of the fluctuating variables.

A Streamline Upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations Under Optimal Control

Abstract In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the Péclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov–Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an optimize-then-discretize approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.

Propagation instabilities of the oblique detonation wave in partially prevaporized n-heptane sprays

Two-dimensional oblique detonation wave (ODW) propagations in partially prevaporized n-heptane sprays are numerically simulated with a skeletal chemical mechanism. The influences of the droplet diameter and total equivalence on oblique detonation are considered. The initiation length is found to increase first and then decrease with increasing initial droplet diameter, and the effect of droplet size is maximized when the initial droplet diameter is approximately $10\ \mathrm {\mu } {\rm m}$ . As the initial droplet diameter varies, unsteady and steady ODWs are observed. In the cases of unsteady ODWs, temperature gradients and non-uniform distributions of the reactant mixture due to droplet evaporation lead to formation of unsteady detonation propagation, therefore leading to fluctuations in the initiation length. The fluctuations in initiation length decrease as the pre-evaporation gas equivalence ratio increases for the unsteady cases. The results further suggest that the relationship between the evaporation layer thickness along the streamline and the corresponding theoretical initiation length can be used to identify an unsteady or steady ODW in cases with large droplets that evaporate behind an oblique shock wave or ODW under the effects of different initial droplet diameters.

Computational study of a thin film flow over a topographical feature using phase-field lattice Boltzmann method

The study employs the phase-field lattice Boltzmann method (PFLBM) to explore the dynamics of a thin film flowing over a topographical feature such as a mound or a trench. The mesoscopic nature of PFLBM makes it a suitable technique for problems involving tracking the evolution of a liquid–air interface. PFLBM simulation results are validated with experimental and analytical results confirming the viability of the numerical approach for such problems. The effect of changing the topographical height, aspect ratio, viscosity ratio, and presence of multiple mounds on the film profiles are systematically analyzed. It is observed that a steady-state solution could not be obtained for large height topographical features. The transition from a steady-state interfacial pattern to an unsteady-steady state is found to depend on the width of the topography. Geometry-based condition is employed to deal with the contact points present in the film dynamics beyond rupture. For large contact angles, the unsteady cases result in film rupture and form a continuous array of droplets of equivalent dimensions at a periodic interval. Increasing the aspect ratio reduces the width of the capillary ridge formed above the topographical feature, while the viscosity ratio reduces the maximum height of the ridge. The shapes of the capillary ridges formed over multiple mounds in the flow direction are independent if the separation between the successive mounds is beyond a critical value. This critical value strongly depends on the capillary number and is independent of the dimensions of the mound.

Improvements in the Prediction of Steady and Unsteady Transition and Mixing in Low-Pressure Turbines by Means of Machine-Learnt Closures

Abstract In this article, novel machine-learnt transition and turbulence models are applied to the prediction of boundary-layer transition and wake mixing in a low-pressure turbine (LPT) cascade, including unsteady inflow cases with incoming wakes. A laminar kinetic energy (LKE) transport approach is employed as baseline transition framework. The application of the machine-learnt explicit algebraic Reynolds stress models (EARSMs) takes advantage of a zonal strategy based on a newly developed sensing function that allows automated wake demarcation. This work compares the performance of several approaches that are based on the application of improved transition models without machine learnt EARSMs, baseline transition model with EARSMs trained for improved wake mixing, and new transition and turbulence closures that are simultaneously developed in a fully coupled way and aimed at improving both transition and wake mixing predictions in LPTs. The investigation is carried out on a cascade of industrial footprint, representative of modern LPT bladings. First, the various modeling frameworks are evaluated, without bar wakes (steady conditions), over a wide range of Reynolds number values. Unsteady Reynolds-averaged Navier–Stokes (URANS) analyses have also been carried out to investigate the predictive capabilities of machine-learnt closures in the presence of incoming bar wakes. Reynolds-averaged Navier–Stokes (RANS)/URANS results are scrutinized against large eddy simulation (LES) calculations and experimental data. It will be demonstrated that machine-learnt EARSMs are capable of producing realistic wake mixing when simply applied on top of the baseline transition models. However, the most accurate predictions will be shown to be provided by fully integrated machine-learnt transition/turbulence closures.

Turbulence Measurements Downstream of a Combustor Simulator Designed for Studies on the Combustor–Turbine Interaction

Turbulence intensity impacts the performance of turbine stages and it is an important inlet boundary condition for CFD computations; the knowledge of its value at the turbine inlet is then of paramount importance. In combustor–turbine interaction experimental studies, combustor simulators replace real combustors and allow for the introduction of flow perturbation at the turbine inlet. Therefore, the turbulence intensity of a combustor simulator used in a wide experimental campaign at Politecnico di Milano is characterized using a hot-wire probe in a blow-down wind tunnel, and the results are compared to URANS CFD computations. This combustor simulator can generate a combination of a swirl profile with a steady/unsteady temperature disturbance. In the cold unsteady disturbance case, hot-wire measurements are phase-averaged at the frequency of the injected perturbation. The combustor simulator turbulence intensity is measured at two different axial positions to understand its evolution.

Scaling laws for natural convection boundary layer of a Pr > 1 fluid on a vertical solid surface subject to a sinusoidal temperature in a linearly-stratified ambient fluid

The understanding of the transient behavior of natural convection boundary layer (NCBL) on a heated vertical solid surface under various heating conditions is of fundamental significance and application importance. In this study, scalings for the parameters representing the behavior of unsteady NCBL flow of a linearly-stratified Pr > 1 fluid on a semi-infinite vertical solid surface heated with a time-varying sinusoidal temperature at different development stages are developed with a scaling analysis, in terms of Ra, Pr, s, and fn, which are the Rayleigh number, Prandtl number, stratification number, and frequency of the sinusoidal temperature, respectively. These scalings are validated and quantified with a series of numerical simulations over wide ranges of Ra, Pr, s, and fn. The frequency of the fluctuations experienced by the NCBL behavior at the transitional stage, due to the stratification of the ambient fluid, is also analyzed, and it is shown that the previously obtained scaling for the unsteady NCBL case with the constant heat flux heating condition is basically applicable for the current case, Ra and fn have additional effects as well due to the time-varying nature of the applied temperature.

On modeling arterial blood flow with or without solute transport and in presence of atherosclerosis

<abstract> <p>In this article, we review previous studies of modeling problems for blood flow with or without transport of a solute in a section of arterial blood flow and in the presence of atherosclerosis. Moreover, we review problems of bio-fluid dynamics within the field of biophysics. In most modeling cases, the presence of red blood cells in the plasma is taken into account either by using a two-phase flow approach, where blood plasma is considered as one phase and red blood cells are counted as another phase, or by using a variable viscosity formula that accounts for the amount of hematocrit within the blood. Both analytical and computational methods were implemented to solve the governing equations for blood flow in the presence of solute transport, which, depending on the type of the investigated problem, could contain momentum, mass conservation, and solute concentration, which were mostly subjected to reasonable approximations. The form of atherosclerosis implemented in the modeling system either was either based on the experimental data for an actual human or was due to a reasonable mathematical modeling for both steady and unsteady atherosclerosis cases. For the wall of the artery itself, which is elastic in nature, modeling equations for the displacement of the artery wall has previously been used, even though their effects on the blood flow inside the artery were shown to be rather small. In some cases, thermal effects were also taken into account by including a temperature equation in the investigation. In the case of the presence of solutes in the blood, various blood flow parameters such as blood pressure force, blood speed, and solute transport were mostly determined or approximated for different values of the parameters that could represent hematocrit, solute diffusion, atherosclerosis height, solute reaction, and pulse frequency. In some studies, available experimental results and data were used in the modeling system that resulted in a more realistic outcome for the blood flow parameters, such as blood pressure force and blood flow resistance. Results have been found for variations of blood flow parameters and solute transport when compared to different values of the parameters. Effects that can increase or decrease the blood flow parameters and solute transport in the artery have mostly been determined, with particular applications for further understanding efforts to improve the patients' health care.</p> </abstract>

Modeling dynamic stall of an airfoil with vortex generators using a double‐wake panel model with viscous–inviscid interaction

AbstractVortex generators (VGs) have been widely applied to wind turbines thanks to their potential to increase aerodynamic performance. Due to the complex inflow perceived by a rotor and the proneness to flow separation, VGs on wind turbines usually experience highly unsteady flow. While there are models that exist to simulate the steady effects of VGs, we lack a fast and efficient tool to model the unsteady performance of airfoils equipped with VGs. This paper adopts an unsteady double‐wake panel model with viscous–inviscid interaction developed to simulate a vertical axis turbine in dynamic stall, adding the capability of predicting the dynamic aerodynamic performance of VG‐equipped airfoils. The results of a series of steady and unsteady cases of an airfoil with different VG configurations in various pitch motions in free and forced transition are verified against experimental data. Results show that the double wake model offers results with sufficient accuracy compared with experimental data to claim the model's validity in a preliminary evaluation of an airfoil's capability to prevent stall with VGs. A few limitations, including the accuracy in prediction the transition location, separation, and reattachment, have been identified for future development.

An R-adaptive algorithm based on self-organizing maps for solving incompressible flows with high-order discontinuous Galerkin methods

Mesh quality is critical for the numerical accuracy of CFD (Computational Fluid Dynamics). Although various techniques have been developed to improve mesh applicability to complex flows, r-adaptive methods have received far too little attention. This paper introduces an r-adaptive algorithm based on the self-organizing maps (SOM) of Kohonen and applies it to unsteady CFD applications using the Discontinuous Galerkin (DG) method. Inspired by the properties of the DG method, the numerical discontinuity over the element interface is quantified as an adaptation indicator. The method requires only one adaptation, followed by solution projection onto the new mesh to save computational costs in unsteady cases. A significant advantage of this method is its higher efficiency without solving complex equations during the adaptation. The performance of the developed mesh adaptation algorithm is tested on the steady laminar viscous flow past a circular cylinder at Re=40 and the unsteady laminar viscous flow past a circular cylinder at Re=100 and two circular cylinders in a side-by-side arrangement at Re=200. Significant improvements in flow contours and coefficients are obtained after adaptation.

On the characteristics of the wake of a wind turbine undergoing large motions caused by a floating structure: an insight based on experiments and multi-fidelity simulations from the OC6 project Phase III

Abstract. This study reports the results of the second round of analyses of the Offshore Code Comparison, Collaboration, Continued, with Correlation and unCertainty (OC6) project Phase III. While the first round investigated rotor aerodynamic loading, here, focus is given to the wake behavior of a floating wind turbine under large motion. Wind tunnel experimental data from the UNsteady Aerodynamics for FLOating Wind (UNAFLOW) project are compared with the results of simulations provided by participants with methods and codes of different levels of fidelity. The effect of platform motion on both the near and the far wake is investigated. More specifically, the behavior of tip vortices in the near wake is evaluated through multiple metrics, such as streamwise position, core radius, convection velocity, and circulation. Additionally, the onset of velocity oscillations in the far wake is analyzed because this can have a negative effect on stability and loading of downstream rotors. Results in the near wake for unsteady cases confirm that simulations and experiments tend to diverge from the expected linearized quasi-steady behavior when the rotor reduced frequency increases over 0.5. Additionally, differences across the simulations become significant, suggesting that further efforts are required to tune the currently available methodologies in order to correctly evaluate the aerodynamic response of a floating wind turbine in unsteady conditions. Regarding the far wake, it is seen that, in some conditions, numerical methods overpredict the impact of platform motion on the velocity fluctuations. Moreover, results suggest that the effect of platform motion on the far wake, differently from original expectations about a faster wake recovery in a floating wind turbine, seems to be limited or even oriented to the generation of a wake less prone to dissipation.

Impact of Unsteady Wakes on the Secondary Flows of a High-Speed Low-Pressure Turbine Cascade

The aerodynamics of a high-speed low-pressure turbine (LPT) cascade were investigated under steady and unsteady inlet flows. The tests were performed at outlet Mach (M) and Reynolds numbers (Re) of 0.90 and 70k, respectively. Unsteady wakes were simulated by means of a wake generator equipped with bars. A bar reduced frequency (f+) of ∼0.95 was used for the unsteady case. The inlet flow field was characterized in terms of the total pressure profile and incidence. The blade aerodynamics at midspan and the secondary flow region were investigated by means of pneumatic taps and hot-film sensors. The latter provided a novel view into the impact of the secondary flows on the heat transfer topology on the blade suction side (SS). The cascade performance was quantified in terms of the outlet flow angle and losses by means of a directional multi-hole probe. The results report the phase-averaged impact of unsteady wakes on the secondary flow structures in an open test case high-speed LPT geometry.