Unsteady Potential Flow Research Articles

Prandtl-Meyer Reflection Configurations, Transonic Shocks, and Free Boundary Problems

We are concerned with the Prandtl-Meyer reflection configurations of unsteady global solutions for supersonic flow impinging upon a symmetric solid wedge. Prandtl (1936) first employed the shock polar analysis to show that there are two possible steady configurations: the steady weak shock solution and the steady strong shock solution, when a steady supersonic flow impinges upon the solid wedge – the half-angle of which is less than a critical angle (i.e., the detachment angle), and then conjectured that the steady weak shock solution is physically admissible since it is the one observed experimentally. The fundamental issue of whether one or both of the steady weak and strong shocks are physically admissible has been vigorously debated over the past eight decades and has not yet been settled in a definitive manner. On the other hand, the Prandtl-Meyer reflection configurations are core configurations in the structure of global entropy solutions of the two-dimensional Riemann problem, while the Riemann solutions themselves are local building blocks and determine local structures, global attractors, and large-time asymptotic states of general entropy solutions of multidimensional hyperbolic systems of conservation laws. In this sense, we have to understand the reflection configurations in order to understand fully the global entropy solutions of two-dimensional hyperbolic systems of conservation laws, including the admissibility issue for the entropy solutions. In this monograph, we address this longstanding open issue and present our analysis to establish the stability theorem for the steady weak shock solutions as the long-time asymptotics of the Prandtl-Meyer reflection configurations for unsteady potential flow for all the physical parameters up to the detachment angle. To achieve these, we first reformulate the problem as a free boundary problem involving transonic shocks and then obtain appropriate monotonicity properties and uniform a priori estimates for admissible solutions, which allow us to employ the Leray-Schauder degree argument to complete the theory for all the physical parameters up to the detachment angle.

The effect of variable rotary motion of a flapping wing rotor on its aerodynamic performance based on an improved unsteady vortex ring method

In the previous aerodynamic analysis using a potential flow-based unsteady aerodynamic model for the flapping wing rotor (FWR), the average rotary velocity and flapping frequency together with the flapping and twist angles are usually taken as the FWR kinematics of motion. In the present study, an unsteady vortex ring method (UVRM) is developed to take the rotary velocity variation of the FWR during the flapping motion into account based on 3D unsteady potential flow theory. The UVRM is validated by comparing with the published results of a flapping wing and an FWR model. An FWR test model and the corresponding experimental platforms including a wing motion tracking system and a force measurement system are built to measure the FWR motion and associated forces. Three FWR test cases of different input voltages and kinematics of flapping motion are considered to measure the aerodynamic forces and compare with the UVRM results. The results show that the differences between the measured and pre-set flapping angles (-50°∼ 20°) for the FWR are negligible for all the cases, but the differences in the variation amplitudes of the measured twist angle and the corresponding pre-set rigid twist angles can be as much as about 15°. Also, the FWR rotary velocity varies dramatically during a flapping cycle. For one of the cases with the -10°∼30° pre-set twist angle and 4 V input voltage, the difference between the maximum and minimum rotary speed is about 1451°/s, which makes a significant effect on the FWR aerodynamic performance. The results also show that the calculated lift forces using the UVRM and the measured real-time rotary speeds during a flapping motion are very close to the experimental results. While the calculated results by taking the FWR's average rotary speeds show apparent differences (up to about 25%) from the measured lift forces in most of the cases. Overall, the UVRM provides an efficient method for the FWR aerodynamic analysis with higher accuracy by taking the effect of variable rotary motion and twist angle of the FWR during a flapping cycle into account.

Flutter and Limit-Cycle Oscillations of a Panel Using Unsteady Potential Flow Aerodynamics

The potential aerodynamic theory is implemented to model the flow over a flexible panel. The present study compares the more complete unsteady version of the potential flow theory with its simplification known as the linear piston theory for different Mach numbers and the two- and three-dimensional aerodynamic models. The piston theory is “local” in space and time because it assumes the pressure at a spatial point and time only depends on the panel deformation at the same point and time. The full potential flow model includes the effect of the past history of the panel deformation and the spatial distribution of the panel deformation on the pressure at any instant in time and at any point in space. Aeroelastic analysis is made to trace the flutter onset critical condition based on the limit cycle oscillation amplitudes, and the results are compared with the more traditional implementation using piston theory. Conclusions are made based on the use and application of this more complete aerodynamic theory, particularly for near-transonic and hypersonic flow regimes. Subsonic results are also presented in this study.

Semi-Analytical Methods in the Problem of Deformation of a Fluid Strip

A problem from a class of unsteady plane potential flows with a free boundary is considered. The entire boundary occupied by the liquid is free, and a zero pressure is maintained. There are neither external nor capillary forces. The motion is driven by inertia. The parameters prescribed at the initial time are the velocity field and the domain occupied by the fluid. The task is to determine these parameters at subsequent time instants. The solution is sought in the form of power series, which are then summed up with the use of the Pade approximation.

Analysis of Interaction between Leakage Flow and Upstream Wake by Proper Orthogonal Decomposition Applied

The turbine blade generally works in severe aerodynamic environments such as high temperature and high pressure. Various forms of secondary flow, unsteady potential flow, periodic wake in the blade passage, and strong interactions among these flow structures affect the efficiency of the turbine seriously. This paper performs a numerical simulation of the T106 cascade to investigate the flow characteristics and performance under the interaction between upstream wake and leakage flow. The cascade flow with and without upstream wake are decomposed by proper orthogonal decomposition, and the mode characteristics of leakage flow and wake are obtained. The results indicate that periodic wake can significantly reduce the energy loss in the channel and tip leakage flow has a greater impact in the region near the shroud plane. According to the proper orthogonal decomposition, the second to seventh modes account for 96.14% of the total energy except for the time-averaged mode. The second mode is identified as the wake mode and leakage flow can significantly increase vorticity in the tip region. The harmonics of the wake mode are greatly affected by leakage flow, especially at the trailing edge region.

Speed and delivered power in waves — Predictions with CFD simulations at full scale

An efficient numerical method is proposed to estimate delivered power and speed loss for a ship in wind and waves. The added resistance in waves, obtained with an unsteady potential flow panel method, is added to the calm water resistance from a steady-state potential flow/RANS method coupled with a body force propeller model for self-propulsion. A comparison of numerical and experimental results is made for added resistance, calm water resistance and delivered power. A good agreement is obtained. As a practical application, the approach is used to calculate the weather factor, fw, of the Energy Efficiency Design Index (EEDI). The calculated weather factor is consistent with the values derived from full-scale measurements included in a database of similar ships.

Experimental mitigation of large-amplitude transverse gusts via closed-loop pitch control

In this work, we experimentally demonstrate the utility of unsteady potential flow models in developing closed-loop control strategies for mitigating the lift transients on a wing experiencing large-amplitude transverse gusts. The developed closed-loop controller mitigates lift for gusts of various strengths and directions and for wings with pre- and post-stall angles of attack. Time-resolved force and flow field measurements are used to discover the salient flow physics during these encounters and illustrate how closed-loop actuation mitigates their lift transients.

Persistence of the steady planar normal shock structure in 3‐D unsteady potential flows

AbstractThis paper concerns the dynamic stability of the steady three‐dimensional (3‐D) wave structure of a planar normal shock front intersecting perpendicularly to a planar solid wall for unsteady potential flows. The stability problem can be formulated as a free boundary problem of a quasi‐linear hyperbolic equation of second order in a dihedral‐space domain between the shock front and the solid wall. The key difficulty is brought by the edge singularity of the space domain, the intersection curve between the shock front and the solid wall. Different from the two‐dimensional (2‐D) case, for which the singular part of the boundary is only a point, it is a curve for the 3‐D case in this paper. This difference brings new difficulties to the mathematical analysis of the stability problem. A modified partial hodograph transformation is introduced such that the extension technique developed for the 2‐D case can be employed to establish the well‐posed theory for the initial‐boundary value problem of the linearized hyperbolic equation of second order in a dihedral‐space domain. Moreover, the extension technique is improved in this paper such that loss of regularity in the a priori estimates on the shock front does not occur. Thus, the classical nonlinear iteration scheme can be constructed to prove the existence of the solution to the stability problem, which shows the dynamic stability of the steady planar normal shock without applying the Nash–Moser iteration method.

Evolution of nonlinear perturbations for a fluid flow with a free boundary. Exact results

The problem of a plane unsteady potential flow of an ideal incompressible fluid bounded by free boundary segments with a constant pressure and by solid walls moving in accordance with a known law is considered. External forces are absent, and capillary forces are neglected. An approach to constructing exact solutions for this type of problem is proposed. The corresponding solutions can be treated as nonlinear perturbations of a certain base flow. As an example of the application of this approach, nonlinear perturbations in a known problem of a fluid flow with a linear velocity field in the region bounded by a straight-line free boundary and parallel approaching or receding solid walls are considered. It is demonstrated that perturbations grow, which leads to variants of the formation of singularities on the free surface of the fluid within a finite time: formation of droplets, bubbles or cusps. A solution describing the collapse of a bubble in a fluid layer bounded by two approaching solid walls has also been found and studied. Thus, a new method of studying nonlinear stability of complicated unsteady fluid flows with combined boundary conditions is proposed and tested.

Local well-posedness of unsteady potential flows near a space corner of right angle

In this paper we are concerned with the local well-posedness of the unsteady potential flows near a space corner of right angle, which could be formulated as an initial-boundary value problem of a hyperbolic equation of second order in a cornered-space domain. The corner singularity is the key difficulty in establishing the local well-posedness of the problem. Moreover, the boundary conditions on both edges of the corner angle are of Neumann-type and fail to satisfy the linear stability condition, which makes it more difficult to establish a priori estimates on the boundary terms in the analysis. In this paper, extension methods will be updated to deal with the corner singularity, and, based on a key observation that the boundary operators are co-normal, new techniques will be developed to control the boundary terms.

A Study of the Effects of Tornado Translation on Wind Loading Using a Potential Flow Model

This paper investigates the effects of tornado translation on pressure and overall force experienced by an airfoil subjected to tornado loading and presents a framework to reproduce the flow conditions and effects of a moving tornado. A thin symmetrical airfoil was used to explore the effects of tornado translation on a body. A panel method was used to compute the flow around an airfoil and an idealised tornado is represented using a moving vortex via unsteady potential flow. Analysis showed that the maximum overall pressure at a point was found to increase by up to 20% when the normalised translating velocity was 10% of the tangential velocity, but increases up to 60% when the normalised translating velocity is 30% of the tangential velocity. Investigation on the impact of varying airfoil thickness (Case 2) revealed that the location of the tornado has significant effect on the overall lift force. However, the overall lift force appeared to be largely insensitive to the tornado translation velocity due gross changes in pressure on either side of the airfoil cancelling each other out. Further comparison with varying airfoil sizes and distance to tornado translating path (Case 3) showed that the relative inflow and outflow angle is the primary factor affecting the lift on the airfoil. Additionally, the maximum forces on a body subjected to a moving tornado can be predicted using uniform flow providing that the appropriate range of inflow angles are known. Based on the analysis on the database of National Oceanic and Atmospheric Administration (NOAA), the normalised translation speed of the recorded tornadoes across the EF scales, appears to vary from 0.25 to 0.37, with an average of 0.32 (∼18.8 m/s). Finally, the framework using uniform flow to reproduce the flow conditions which are comparable to those generated by a translating vortex simulator is proposed and discussed in detail.

Transonic small disturbance unsteady potential flow over very high aspect ratio wings

AbstractIn this paper, the prediction of the unsteady flow field over typical high aspect ratio (AR) wings in the transonic flow regime but below the sonic Mach number is of interest. The methodology adopted is a computational approach based on the transonic small disturbance unsteady potential equation. It is shown that the higher AR wings generally have a higher lift coefficient as well as a higher lift-to-drag ratio. With NASA’s common research model (CRM) wing, there is an increase in maximum lift with increasing AR while the induced drag is almost the same. There is also an optimum sweep angle, which is different for each angle-of-attack so that variable sweep lifting surfaces may be designed to provide optimum solutions. The computed flutter speeds indicate an expected reduction with increasing AR.

Scaling the dynamic response and stability of composite hydrodynamic lifting surfaces

Imperfectly scaled models are commonplace in aerodynamics and hydrodynamics because few facilities can meet all the flow and structural scaling requirements. The objectives of this work are to (1) derive and numerically validate scaling relations for the steady-state and dynamic hydroelastic response and stability of hydrodynamic composite lifting surfaces and (2) to investigate the effect of imperfect scaling on the steady-state and dynamic response, as well as hydroelastic stability of said surfaces. We derive proper scaling laws and validate them through numerical simulations using unsteady potential flow and composite beam theory. We investigate the prediction of inception of a new, low-frequency mode that only develops at sufficiently high speeds using scaled models. We demonstrate scaling effects on the modal frequencies, damping coefficients, and hydroelastic instability boundaries by comparing cases with fully enforced scaling laws and cases with relaxed similarities. Using the same material for both the prototype and reduced length-scale models results in imperfect scaling; the steady-state and dynamic hydroelastic responses are not scaled, and smaller length-scale models exhibit delayed hydroelastic instabilities. The conclusion is that using appropriate materials for model scale tests to satisfy Cauchy number similarity is necessary for similar steady-state response, but dynamic similarity additionally requires similar solid-to-fluid added mass ratio. Proper scaling allows one to accurately interpret and assess the performance and stability of full-scale flexible composite lifting surfaces such as foils, propulsors, turbines, fins, rudders, energy saving devices, and energy harvesters.

Dynamics of an unbalanced circular foil and point vortices in an ideal fluid

This paper addresses the problem of the motion of an unbalanced circular foil and point vortices in an ideal incompressible fluid. Using Bernoulli's theorem for unsteady potential flow, the force due to the pressure from the fluid on the foil is obtained for an arbitrary vortex motion. A detailed analysis is made of the case of free vortex motion in which a Hamiltonian reduction by symmetries is performed. For the resulting system, relative equilibria corresponding to the motion of an unbalanced foil and a vortex in a circle or in a straight line are found and their stability is investigated. New examples of stationary configurations of a vortex and a foil are given. Using a Poincaré map, it is also shown that in the general case of an unbalanced circular foil the reduced system exhibits chaotic trajectories.

An integrated measurement approach for the determination of the aerodynamic loads and structural motion for unsteady airfoils

The structural motion and unsteady aerodynamic loads of a pitching airfoil model that features an actuated trailing edge flap are determined experimentally using a single measurement and data processing system. This integrated approach provides an alternative to the coordinated use of multiple measurement systems for simultaneous position and flow field measurements in large-scale fluid–structure interaction experiments. The measurements in this study are performed with a robotic PIV system using Lagrangian particle tracking. Flow field measurements are obtained by seeding the flow with helium-filled soap bubbles, while the structural measurements are performed by tracking fiducial markers on the model surface. The unsteady position and flap deflection of the airfoil model are determined from the marker tracking data by fitting a rigid body model, that accounts for the motion degrees of freedom of the airfoil model, to the measurements. For the determination of the unsteady aerodynamic loads (lift and pitching moment) from the flow field measurements, two different approaches are evaluated, that are both based on unsteady potential flow and thin airfoil theory. These methods facilitate an efficient non-intrusive load determination on unsteady airfoils and produce results that are in good agreement with reference measurements from pressure transducers.

Added resistance, heave and pitch for the KVLCC2 tanker using a fully nonlinear unsteady potential flow boundary element method

In this paper, a fully nonlinear unsteady potential flow method is used to predict added resistance, heave and pitch for the KVLCC2 hull in regular head waves at design speed. The method presents a nonlinear decomposition of the velocity potential and the wave field and an adaptive grid refinement. A formulation for the acceleration potential is used to obtain the pressure. To improve computational efficiency, a Barnes-Hut algorithm is introduced. A grid dependency study and a study on the impact of different time steps on the solution are performed. Numerical results have been compared with experimental data for the design speed. A general good agreement is found for added resistance, especially for longer waves. Heave and pitch are properly computed for all wave lengths in the range λ/Lpp=0.4 to 1.4.

Time domain TEBEM method of ship motion in waves with forward speed by using impulse response function formulation

A numerical method for solving 3D unsteady potential flow problem of ship advancing in waves is proposed. The flow field is divided into inner and outer domain by an artificial matching surface. The inner domain includes ship-wetted surface, matching surface and free surface inside. The free-surface condition is formulated by perturbation about the double-body flow (DB model) or uniform incoming flow assumption (NK model). The outer domain contains matching surface, free surface outside and infinite far-field boundary, where only NK model is applied. The simple Green function and transient free surface Green function are used to form the boundary integral equation (BIE) by the impulse response function method for the inner and outer domain. Taylor Expansion Boundary Element Method (TEBEM) is utilized to solve the BIEs. Matching conditions for the inner-domain and outer-domain flow are enforced by the continuity of velocity potential and normal velocity on the matching surface. In this paper, several ships are selected to predict the hydrodynamic coefficients and displacement at different forward speed. The experiment measurements in physical tank is also used to validate the accuracy of numerical results. Good agreement of 6DOF ship motions in various wave headings can be obtained using TEBEM method.

Unsteady propulsion and the acoustic signature of undulatory swimmers in and out of ground effect

Propulsive performance and acoustic emission of undulatory swimmers are investigated using an integrated unsteady potential flow and acoustic boundary element solver. Anguilliform and carangiform swimming gaits are modeled by a deforming NACA 0012 airfoil section. The effect of a ground plane on undulatory swimming and associated noise generation is examined as a function of reduced frequency and above plane altitude with the method of images. Ground effect becomes pronounced when swimming within half of a chord length from the ground plane, where the thrust, power, propulsive efficiency, and peak acoustic pressure all increase for both gaits.

Time-domain TEBEM method for mean drift force and moment of ships with forward speed under the oblique seas

A numerical method for solving 3D unsteady potential flow problem of ship advancing in waves is put forward. The flow field is divided into an inner and an outer domain by introducing an artificial matching surface. The inner domain is surrounded by ship wetted surface and matching surface as well as part of the free surface. The free surface condition for the inner domain is formulated by perturbation about the double-body flow or uniform incoming flow assumption. The outer domain is surrounded by matching surface and the rest free surface as well as infinite far-field radiation boundary. The free surface condition for the outer domain is formulated by perturbation about uniform incoming flow. The simple Green function and transient free surface Green function are used to form the boundary integral equation (BIE) for the inner and outer domains, respectively. Taylor Expansion Boundary Element Method (TEBEM) is utilized to solve the double-body flow and inner domain and outer domain unsteady flow BIE. Matching conditions for the inner domain flow and outer domain flow are enforced by the continuity of velocity potential and normal velocity on the matching surface. Direct pressure integration on ship wetted surface is used to obtain the first-order and second-order wave forces (moments). The numerical predictions on the displacement, added resistance, sway mean drift force and yaw mean drift moment of the modified KVLCC2 ship at different forward speeds are investigated by the proposed TEBEM method. It is also compared with the other numerical results. The physical tank experiment results are also developed to validate the accuracy of numerical tank results. Compared with the experiment solutions, a good agreement can be obtained by TEBEM method.

Non-intrusive determination of the unsteady surface pressure and aerodynamic loads on a pitching airfoil

The unsteady surface pressure distribution and aerodynamic loads on a pitching airfoil are determined non-intrusively using PIV measurements. An experimental test case is considered where the flow around the airfoil is mostly attached while the unsteady effects on the aerodynamic loads are significant. The surface pressure is calculated from the flow velocity measurements in the vicinity of the airfoil surface, that are obtained with a robotic PIV system, by using relations from unsteady potential flow and thin airfoil theory. The proposed approach is a robust and computationally efficient approach to obtain non-intrusive measurements of the unsteady surface pressure distribution and the aerodynamic loads, that are in good agreement with reference data from installed pressure transducer sensors.