Unsteady Potential Flow Research Articles

Study on Mooring Design and Calculation Method of Ocean Farm Based on Time-Domain Potential Flow Theory

In order to calculate the mooring force of a new semi-submerged Ocean Farm quickly and accurately, based on the unsteady time-domain potential flow theory and combined the catenary model, the control equation of mooring cable is established, and the mooring force of the platform under the wave spectrum is calculated. First of all, based on the actual situation of the ocean environment and platform, the mooring design of the platform is carried out, and the failure analysis and sensitivity analysis of the single anchor chain by the time domain coupling method are adopted: including different water depth, cycle, pretension size, anchor chain layout direction and wind speed, etc. The analysis results confirm the reliability of anchoring method. Based on this, the mooring point location of the platform is determined, the force of each anchor chain in the anchoring process is calculated, and the mooring force and the number of mooring cables are obtained for each cable that satisfies the specification, the results of this paper can provide theoretical calculation methods for mooring setting and mooring force calculation of similar offshore platforms.

Time-domain TEBEM method for wave added resistance of ships with forward speed

A numerical method for solving 3D unsteady potential flow problem of ship advancing in waves was put forward. The flow field was divided into an inner and an outer domain by introducing an artificial matching surface. The inner domain was surrounded by a ship-wetted surface and a matching surface as well as part of the free surface. The free-surface condition for the inner domain was formulated by perturbation about the double-body (DB) flow assumption. The outer domain was surrounded by a matching surface and the rest had a free surface as well as an infinite far-field radiation boundary. The free-surface condition for the outer domain was formulated by perturbation of the uniform incoming flow. The simple Green function and transient free-surface Green function were used to form the boundary integral equation (BIE) for the inner and outer domains, respectively. The Taylor expansion boundary element method (TEBEM) was adopted to solve the DB flow and inner-domain and outer-domain unsteady flow BIE. Matching conditions for the inner-domain flow and outer-domain flow were enforced by the continuity of velocity potential and normal velocity on the matching surface. Direct pressure integration on the ship-wetted surface was applied to obtain the first- and second-order wave forces. The numerical prediction on the displacement, acceleration and added resistance of the 14000-TEU container ship at different forward speeds were investigated by the proposed TEBEM method. Reynolds-Averaged Navier–Stokes (RANS) equations based on Computational Fluid Dynamics (CFD) method were adopted to compare with TEBEM method. The physical tank experiment results also validated the accuracy of the numerical tank results. Compared with the experimental solutions, TEBEM obtained good agreement with the RANS CFD method. TEBEM, however, was much more efficient and robust.

Flow Interactions Between Low Aspect Ratio Hydrofoils in In-line and Staggered Arrangements.

Many species of fish gather in dense collectives or schools where there are significant flow interactions from their shed wakes. Commonly, these swimmers shed a classic reverse von Kármán wake, however, schooling eels produce a bifurcated wake topology with two vortex rings shed per oscillation cycle. To examine the schooling interactions of a hydrofoil with a bifurcated wake topology, we present tomographic particle image velocimetry (tomo PIV) measurements of the flow interactions and direct force measurements of the performance of two low-aspect-ratio hydrofoils () in an in-line and a staggered arrangement. Surprisingly, when the leader and follower are interacting in either arrangement there are only minor alterations to the flowfields beyond the superposition of the flowfields produced by the isolated leader and follower. Motivated by this finding, Garrick’s linear theory, a linear unsteady hydrofoil theory based on a potential flow assumption, was adapted to predict the lift and thrust performance of the follower. Here, the follower hydrofoil interacting with the leader’s wake is considered as the superposition of an isolated pitching foil with a time-varying cross-stream velocity derived from the wake flow measurements of the isolated leader. Linear theory predictions accurately capture the time-averaged lift force and some of the major peaks in thrust derived from the follower interacting with the leader’s wake in a staggered arrangement. The thrust peaks that are not predicted by linear theory are likely driven by spatial variations in the flowfield acting on the follower or nonlinear flow interactions; neither of which are accounted for in the simple theory. This suggests that unsteady potential flow theory that does account for spatial variations in the flowfield acting on a hydrofoil can provide a relatively simple framework to understand and model the flow interactions that occur in schooling fish. Additionally, schooling eels can derive thrust and efficiency increases of 63-80% in either a in-line or a staggered arrangement where the follower is between two branched momentum jets or with one momentum jet branch directly impinging on it, respectively.

Integration of a deep fluid equation with a free surface

We show that the Euler equations describing the unsteady potential flow of a two-dimensional deep fluid with a free surface in the absence of gravity and surface tension can be integrated exactly under a special choice of boundary conditions at infinity. We assume that the fluid surface at infinity is unperturbed, while the velocity increase is proportional to distance and inversely proportional to time. This means that the fluid is compressed according to a self-similar law. We consider perturbations of a self-similarly compressible fluid and show that their evolution can be accurately described analytically after a conformal map of the fluid surface to the lower half-plane and the introduction of two arbitrary functions analytic in this half-plane. If one of these functions is equal to zero, then the solution can be written explicitly. In the general case, the solution appears to be a rapidly converging series whose terms can be calculated using recurrence relations.

Aeroelastic Reduced Order Model: Kriging-Corrected Potential Flow

A novel aeroelastic reduced order model is developed for two-dimensional gust interaction modeling in transonic flows. Such conditions exhibit a key nonlinearity: sonic shock motion. The proposed approach comprises an aerodynamics reduced order model coupled to a structural model. The aerodynamics model consists of corrected integrated force coefficients computed by an unsteady potential flow method (incompressible vortex panels). The corrections are obtained from kriging surrogates trained using inviscid compressible computational fluid dynamics data. Two gust lengths (namely, long and short) are simulated for this purpose and serve as the upper and lower bounds of the model. The developed aeroelastic reduced order model is evaluated via modeling untried gust cases, the lengths of which fall between the training cases. Applied to the FFAST Crank and NACA0012 airfoils, the approach accurately reproduces aeroelastic response—with deviations never exceeding 8.0 and 10.5%, respectively. In terms of computational cost, the trained models were more than an order of magnitude faster than the computational fluid dynamics simulations.

Efficient prediction of classical flutter stability of turbomachinery blade using the boundary element type numerical method

In this paper classical flutter phenomena in power turbine rotor is studied. A medium fidelity 2D flow solver based on boundary element method is developed for this purpose. The classical flutter parameters in turbomachinery cascades such as aerodynamic damping at different inter-blade phase angle are estimated using flutter stability analysis flow solver. The flow solver is developed using 2D unsteady potential flow based panel method. The unsteady aerodynamic loading on the vibrating cascade is estimated using a traveling-wave mode oscillation. The simulated aerodynamic damping and pressure coefficient using boundary element method are compared against both experimental data and the computational fluid dynamics model’s results at different flow conditions. The boundary element based method results demonstrate good agreement with experimental data. The boundary element based flow solver shows significant reduction in computational time compared to computational fluid dynamics model.

Unsteady response of airfoils due to small-scale pitching motion with considerations for foil thickness and wake motion

Unsteady pressures, forces, and pitching moments generated by foils experiencing vibratory motion in an incompressible, attached flow configuration are studied within this work. Specifically, two-dimensional, unsteady potential flow and unsteady Reynolds-Averaged Navier–Stokes calculations are performed on various Joukowski foils undergoing sinusoidal, variable amplitude, small-scale pitching motion at a chord-based Reynolds number of 106 over a range of reduced frequencies between 0.01–100. These calculated results from both approaches are compared directly to predictions from implementing the Theodorsen model, which treats foils as infinitely thin, flat plates that shed a planar sheet of vorticity. The effects of relaxing these seemingly strict conditions are explored, and the particular terms which control the unsteady responses are identified and discussed. For increasing pitch amplitudes and reduced frequencies the shed wake is seen to become quite non-planar and to form coherent vortex structures. Despite this wake behavior, the normalized airfoil responses at the disturbance reduced frequency are seen to be largely unaffected. However, non-negligible responses are generated across a wide range of other frequencies. Potential flow calculations for symmetric Joukowski foils show that there is marginal effect of foil thickness at reduced frequencies less than one. For higher reduced frequency conditions however, the unsteady lift response is seen to experience both an amplification of level and a phase shift relative to the Theodorsen model. A specific augmenting expression is developed for this behavior through analysis within the potential flow framework.

On the leading-edge suction and stagnation-point location in unsteady flows past thin aerofoils

Unsteady thin-aerofoil theory is a low-order method for calculating the forces and moment developed on a camber line undergoing arbitrary motion, based on potential-flow theory. The vorticity distribution is approximated by a Fourier series, with a special ‘A0’ term that is infinite at the leading edge representing the ‘suction peak’. Though the integrated loads are finite, the pressure and velocity at the leading edge in this method are singular owing to the term. In this article, the principle of matched asymptotic expansions is used to resolve the singularity and obtain a uniformly valid first-order solution. This is performed by considering the unsteady thin-aerofoil theory as an outer solution, unsteady potential flow past a parabola as an inner solution, and by matching them in an intermediate region where both are asymptotically valid. Resolution of the leading-edge singularity allows for derivation of the velocity at the leading edge and location of the stagnation point, which are of physical and theoretical interest. These quantities are seen to depend on only the A0 term in the unsteady vorticity distribution, which may be interpreted as an ‘effective unsteady angle of attack’. The leading-edge velocity is proportional to A0 and inversely proportional to the square root of leading-edge radius, while the chordwise stagnation-point location is proportional to the square of A0 and independent of the leading-edge radius. Closed-form expressions for these in simplified scenarios such as quasi-steady flow and small-amplitude harmonic oscillations are derived.

Persistence of the Steady Normal Shock Structure for the Unsteady Potential Flow

This paper is devoted to the study of the stability of the steady normal shock structure in potential flows under an unsteady perturbation. The dynamic stability problem is formulated as the well-p...

Algorithm for Constructing Exact Solutions of the Problem of Unsteady Plane Motion of a Fluid with a Free Surface

Unsteady plane potential flows of an ideal incompressible fluid with a free surface in the absence of external forces and capillarity are studied. An algorithm to construct exact solutions for such flows has been proposed on the basis of the analysis of compatibility conditions of the equations of motion and an auxiliary complex transport equation. This algorithm makes it possible to significantly expand the list of known exact nontrivial solutions of the considered classical problem. This list has recently consisted of only a few Dirichlet solutions: flows for which the surface of the fluid is a parabola, ellipse, or hyperbola. This algorithm allows reproducing a recently found class of solutions specified by the Hopf equation for the complex velocity and finding a fundamentally new large class of solutions for which flows are described by the Hopf equation for the inverse complex velocity.

Investigation of the unsteady responses of airfoils due to small-scale motion

Unsteady pressures, forces, and pitching moments generated by foils experiencing vibratory motion in an incompressible, attached flow configuration are studied within this work. Specifically, two-dimensional, unsteady potential flow calculations are performed on Joukowski foils of varying thickness undergoing variable amplitude, small-scale, heaving or pitching motion over a range of reduced frequencies between 0.01 and 100. These calculated results are compared directly to predictions from implementing the Theodorsen model, which treats foils as infinitely thin flat plates that shed a planar sheet of vorticity. The effects of relaxing these seemingly strict assumptions of the Theodorsen model are explored, and focus is placed on results which show deviations from the Theodorsen model. These include altered unsteady responses for finitely thick foils and non-zero mean angle of attack conditions, non-linearity of the wake of shed vorticity, and analysis of the unsteady streamwise force. Further, within the potential flow framework the particular terms which control the unsteady responses are identified.

Lagrange crisis and generalized variational principle for 3D unsteady flow

PurposeA three-dimensional (3D) unsteady potential flow might admit a variational principle. The purpose of this paper is to adopt a semi-inverse method to search for the variational formulation from the governing equations.Design/methodology/approachA suitable trial functional with a possible unknown function is constructed, and the identification of the unknown function is given in detail. The Lagrange multiplier method is used to establish a generalized variational principle, but in vain.FindingsSome new variational principles are obtained, and the semi-inverse method can easily overcome the Lagrange crisis.Practical implicationsThe semi-inverse method sheds a promising light on variational theory, and it can replace the Lagrange multiplier method for the establishment of a generalized variational principle. It can be used for the establishment of a variational principle for fractal and fractional calculus.Originality/valueThis paper establishes some new variational principles for the 3D unsteady flow and suggests an effective method to eliminate the Lagrange crisis.

Local singular boundary method for two-dimensional steady and unsteady potential flow

This paper focuses on deriving a local variant of a singular boundary method for steady and unsteady potential flow. Compared to the global variant, local leads to the sparse resulting matrix of the system of equations and thus makes the solution of large-scale tasks more effective. The combination of the singular boundary method and the finite collocation method were used for localization. The article compares the results of global and local variation on several examples for both steady and unsteady flow, and the dependence of the solution precision on the density of the nodal network and the dimensions of the stencil was also studied.

Cumulative jet generation in a plane parallel potential flow of the perfect incompressible fluid

This manuscript deals with the plane parallel unsteady potential flow of the perfect incompressible fluid limited by a free boundary. As the result of the free boundary deformation, cumulative jets may form. A cumulative jet generation in a problem of cylindrical cavity deformation is considered in this manuscript. A numerical algorithm for this problem is proposed. Based on the boundary elements method with no saturation, the algorithm has a high order of approximation and reasonable computation time. The distribution of points along the length of the cavity boundary is required to be constant at computation time. This requirement provides the numerical stability of the algorithm. To verify the correctness of the proposed algorithm, conservation laws are used.

Numerical investigation of flow pattern and hydrodynamic forces of submerged marine propellers using unsteady boundary element method

In this study, a numerical model of unsteady potential flow around submerged marine propellers has been developed. The boundary element approach in combination with time stepping method to model free wake dynamics has been implemented. An important feature of this method in the simulation of pressure-dominant problems is a proper balance between time and accuracy in the numerical process. Another advantage of time stepping method is that there is no need to define wake geometry before modeling. Due to inherent instability of boundary integral equations, a smoothing function to damp the effect of singularities is imposed to the solution. The main innovative idea of this work is that the effect of this mollifier function with different constants for elimination of singularity in mean induced velocity equation has been investigated. The mathematical formulation and implementation of numerical algorithm followed by a number of test cases to verify the model are presented. The results indicate good agreements between model and experiments.

Stability and Control of Tailless Aircraft Using Variable-Fidelity Aerodynamic Analysis

The purpose of this study is to evaluate the stability and control characteristics of the tailless aircraft using a hierarchy of variable-fidelity aerodynamics analysis methods, including unsteady linear potential flow, Euler, and Reynolds-averaged Navier–Stokes solvers as low-, medium-, and high-fidelity analysis, respectively. Contributions of the study are numerous. First, derivations of the equations of motion for the tailless aircraft and stability criteria using system matrices in longitudinal and lateral motions were carried out. Second, an efficient time-spectral Reynolds-averaged Navier–Stokes computational fluid dynamics method, which is based on the solution approximation of a discrete Fourier series, and the steady form of a corresponding adjoint solution approach can compute dynamic derivatives at computational cost considerably less than that of the conventional time-marching method without deterioration in solution accuracy. The innovative control effector 101 configuration was chosen, and the validation of the variable-fidelity computation results was made with wind tunnel data for forces, moments, and their derivative values in static and dynamic motions. Finally, example flight conditions of trim, climb, and side slip motions were analyzed by the Reynolds-averaged Navier–Stokes solver for the stability analysis. It is concluded that the weak instabilities observed in some of the dynamic modes can be easily controlled by appropriate control inputs.

Wet and Dry Transom Stern Treatment for Unsteady and Nonlinear Potential Flow Model for Naval Hydrodynamics Simulations

We present a model for the fast evaluation of the total drag of ship hulls operating in both wet and dry transom stern conditions, in calm or wavy water, based on the combination of an unsteady semi-Lagrangian potential flow formulation with fully nonlinear free-surface treatment, experimental correlations, and simplified viscous drag modeling. The implementation is entirely based on open source libraries. The spatial discretization is solved using a streamline upwind Petrov-Galerkin stabilization of an iso-parametric, collocation based, boundary element method, implemented using the open source library deal. II. The resulting nonlinear differential-algebraic system is integrated in time using implicit backward differentiation formulas, implemented in the open source library SUNDIALS. The Open CASCADE library is used to interface the model directly with computer-aided design data structures. The model accounts automatically for hulls with a transom stern, both in wet and dry regimes, by using a specific treatment of the free-surface nodes on the stern edge that automatically detects when the hull advances at low speeds. In this case, the transom stern is partially immersed, and a pressure patch is applied on the water surface detaching from the transom stern, to recover the gravity effect of the recirculating water on the underlying irrotational flow domain. The parameters of the model used to impose the pressure patch are approximated from experimental relations found in the literature. The test cases considered are those of the U.S. Navy Combatant DTMB-5415 and the National Physical Laboratory hull. Comparisons with experimental data on quasi-steady test cases for both water elevation and total hull drag are presented and discussed. The quality of the results obtained on quasi-steady simulations suggests that this model can represent a promising alternative to current unsteady solvers for simulations with Froude numbers below 0.35.

Wet and Dry Transom Stern Treatment for Unsteady and Nonlinear Potential Flow Model for Naval Hydrodynamics Simulations

Wet and Dry Transom Stern Treatment for Unsteady and Nonlinear Potential Flow Model for Naval Hydrodynamics Simulations

Instability of a vortex sheet leaving a right-angled wedge

We examine the dynamics of a semi-infinite vortex sheet attached not to a semi-infinite plate but instead to a rigid right-angled wedge, with the sheet aligned along one of its edges. Our approach to this problem, which was suggested by David Crighton, accords well with the fundamental ethos of Crighton’s work, which was characterized by ‘the application of rigorous mathematical approximations to fluid mechanical idealizations of practically relevant problems’ (Ffowcs Williams, Annu. Rev. Fluid Mech., vol. 34, 2002, pp. 37–49). The resulting linearised unsteady potential flow is forced by an oscillatory dipole in the uniform stream passing along the top of the wedge, while there is stagnant fluid in the remaining quadrant. Spatial instability is considered according to well-established methods: causality is enforced by allowing the frequency to become temporarily complex. The essentially quadrant-type geometry replaces the usual Wiener–Hopf technique by the Mellin transform. The core difficulty is that a first-order difference equation of period 4 requires a solution of period unity. As a result, the complex fourth roots $(\pm 1\pm \text{i})$ of $-4$ appear in the complementary function. The Helmholtz instability wave is excited and requires careful handling to obtain explicit results for the amplitude of the instability wave.

Unsteady Aerodynamic Model Based on the Leading-Edge Stagnation Point

The significance of the leading-edge stagnation point as an unsteady aerodynamic observable is investigated. Using first-order unsteady potential flow theory, a simple model relating the stagnation point to the unsteady lift is derived for an airfoil undergoing arbitrary unsteady maneuvers. In particular, the dynamic circulatory effects are shown to be accounted for by tracking the stagnation point. Linearization permits the superposition of solutions for thickness and camber. The model is validated using numerical simulations, employing both steady and unsteady panel methods. In addition, its validity is investigated experimentally through steady and unsteady wind-tunnel tests of a wing section. The stagnation point is estimated in real time from the convective heat transfer distribution measured using a hot-film sensor array instrumented around the wing’s leading edge. Applications of these findings to flutter suppression and gust-load alleviation are experimentally demonstrated.