Wave Drag Coefficient Research Articles

Multifidelity Comparison of Supersonic Wave Drag Prediction Methods Using Axisymmetric Bodies

Low-fidelity analytic and computational wave drag prediction methods assume linear aerodynamics and small perturbations to the flow. Hence, these methods are typically accurate for only very slender geometries. The present work assesses the accuracy of these methods relative to high-fidelity Euler, compressible computational-fluid-dynamics solutions for a set of axisymmetric geometries with varying radius-to-length ratios (R/L). Grid-resolution studies are included for all computational results to ensure grid-resolved results. Results show that the low-fidelity analytic and computational methods match the Euler CFD predictions to around a single drag count ( ∼1.0×10−4) for geometries with R/L≤0.05 and Mach numbers from 1.1 to 2.0. The difference in predicted wave drag rapidly increases, to over 30 drag counts in some cases, for geometries approaching R/L≈0.1, indicating that the slender-body assumption of linear supersonic theory is violated for larger radius-to-length ratios. All three methods considered predict that the wave drag coefficient is nearly independent of Mach number for the geometries included in this study. Results of the study can be used to validate other numerical models and estimate the error in low-fidelity analytic and computational methods for predicting wave drag of axisymmetric geometries, depending on radius-to-length ratios.

ShipGen: A Diffusion Model for Parametric Ship Hull Generation with Multiple Objectives and Constraints

Ship design is a years-long process that requires balancing complex design trade-offs to create a ship that is efficient and effective. Finding new ways to improve the ship design process could lead to significant cost savings in the time and effort required to design a ship, as well as cost savings in the procurement and operation of a ship. One promising technology is generative artificial intelligence, which has been shown to reduce design cycle times and create novel, high-performing designs. In a literature review, generative artificial intelligence was shown to generate ship hulls; however, ship design is particularly difficult, as the hull of a ship requires the consideration of many objectives. This paper presents a study on the generation of parametric ship hull designs using a parametric diffusion model that considers multiple objectives and constraints for hulls. This denoising diffusion probabilistic model (DDPM) generates the tabular parametric design vectors of a ship hull, which are then constructed into a point cloud and mesh for performance evaluation. In addition to a tabular DDPM, this paper details adding guidance to improve the quality of the generated parametric ship hull designs. By leveraging a classifier to guide sample generation, the DDPM produced feasible parametric ship hulls that maintained the coverage of the initial training dataset of ship hulls with a 99.5% rate, a 149× improvement over random sampling of the design vector parameters across the design space. Parametric ship hulls produced using performance guidance saw an average 91.4% reduction in wave drag coefficients and an average 47.9× relative increase in the total displaced volume of the hulls compared to the mean performance of the hulls in the training dataset. The use of a DDPM to generate parametric ship hulls can reduce design times by generating high-performing hull designs for future analysis. These generated hulls have low drag and high volume, which can reduce the cost of operating a ship and increase its potential to generate revenue.

Investigation of the effect of nose shape and geometry at supersonic speeds for missile performance optimisation

This paper investigated and analysed the influence of the missile nose shapes and geometry on the drag force coefficients at supersonic Mach numbers ranging from 2 to 5. The nose shape investigation includes a pointed cone, pointed ogive, and blunt cone with a constant length and diameter of cylindrical afterbody. Subsequently, the nose length was manipulated to investigate how the nose geometry with different fineness ratios affected each drag component. The drag force coefficient and its components were determined using a program called MATLAB_Aerody, in which a semi-empirical approach was implemented for the calculations. The program was validated against the semi-empirical results with existing theoretical and experimental results. The comparison showed a maximum error of 3.6%. The results of the nose shape analysis show that a sharper and more slender nose with a pointy tip produces a minimal drag. Nose bluntness, however, creates considerable additional drag force due to a higher wave drag coefficient by forming detached or bow shock waves. At supersonic speeds, increasing the fineness ratio significantly reduces the drag force by drastically decreasing the wave drag coefficient with a marginal increase in the skin friction coefficients. However, the results show an insignificant additional gain for the skin friction drag and a reduction in the wave drag coefficient when the nose has a fineness ratio beyond 4.5. Furthermore, the results from both analyses show that the base drag coefficient is invariant with the nose shape and the fineness ratio. Therefore, a better missile performance can be achieved with a sharper and more slender nose shape with a higher fineness ratio.

Barotropic tides in MPAS-Ocean (E3SM V2): impact of ice shelf cavities

Abstract. Oceanic tides are seldom represented in Earth system models (ESMs) owing to the need for high horizontal resolution to accurately represent the associated barotropic waves close to coasts. This paper presents results of tides implemented in the Model for Prediction Across Scales–Ocean or MPAS-Ocean, which is the ocean component within the U.S. Department of Energy developed Energy Exascale Earth System Model (E3SM). MPAS-Ocean circumvents the limitation of low resolution using unstructured global meshing. We are at this stage simulating the largest semidiurnal (M2, S2, N2) and diurnal (K1, O1) tidal constituents in a single-layer version of MPAS-O. First, we show that the tidal constituents calculated using MPAS-Ocean closely agree with the results of the global tidal prediction model TPXO8 when suitably tuned topographic wave drag and bottom drag coefficients are employed. Thereafter, we present the sensitivity of global tidal evolution due to the presence of Antarctic ice shelf cavities. The effect of ice shelves on the amplitude and phase of tidal constituents are presented. Lower values of complex errors (with respect to TPXO8 results) for the M2 tidal constituents are observed when the ice shelf is added in the simulations, with particularly strong improvement in the Southern Ocean. Our work points towards future research with varying Antarctic ice shelf geometries and sea ice coupling that might lead to better comparison and prediction of tides and thus better prediction of sea-level rise and also the future climate variability.

Lift Airfoils Close to the Airfoils in a Flow with the Maximum “Critical” Mach Numbers

The lift airfoils close to the airfoils optimal with respect to the critical Mach numbers М* of two-dimensional bodies optimal with respect to М* are constructed using the direct method. Their almost zero wave drag coefficients сх remain the same not only at the free-stream Mach numbers М0 which are lower than М* but also at М0 perceptibly higher than М*. These new lift airfoils differ from the supercritical lift airfoils whose сх grow extremely rapidly when М0 becomes higher than the designed values. At identical thicknesses and М0 = М* the supercritical lift airfoils implement the greater lift coefficients су. However, due to the difference in the behavior of сх at М0 which are higher than the designed ones, the lift-drag ratio of the supercritical airfoils can become lower for the ratio of су not to сх, but even to the coefficient of total drag.

Wave drag coefficient of nonaxisymmetric irregular-shaped bodies

At high speeds, transonic and supersonic, the wave drag coefficient is a significant part of drag coefficients and it depends on the front surface shape and flow velocity. Drag, wave drag, drag coefficient, and wave drag coefficient for four bodies with the same width, height and length, and different shapes are estimated by CFD (ANSYS Fluent). For different front surface curvatures, at high flow velocity, pressure distribution on the front surface of the body, and flow field as a contour of Mach number are analyzed. The influence of front surface curvature on detached shock wave distance is determined.

Prototype-Scale Physical Model of Wave Attenuation Through a Mangrove Forest of Moderate Cross-Shore Thickness: LiDAR-Based Characterization and Reynolds Scaling for Engineering With Nature

This study investigates the potential of a Rhizophora mangrove forest of moderate cross-shore thickness to attenuate wave heights using an idealized prototype-scale physical model constructed in a 104 m long wave flume. An 18 m long cross-shore transect of an idealized red mangrove forest based on the trunk-prop root system was constructed in the flume. Two cases with forest densities of 0.75 and 0.375 stems/m2 and a third baseline case with no mangroves were considered. LiDAR was used to quantify the projected area per unit height and to estimate the effective diameter of the system. The methodology was accurate to within 2% of the known stem diameters and 10% of the known prop root diameters. Random and regular wave conditions seaward, throughout, and inland of the forest were measured to determine wave height decay rates and drag coefficients for relative water depths ranging 0.36 to 1.44. Wave height decay rates ranged 0.008–0.021 m–1 for the high-density cases and 0.004–0.010 m–1 for the low-density cases and were found to be a function of water depth. Doubling the forest density increased the decay rate by a factor two, consistent with previous studies for other types of emergent vegetation. Drag coefficients ranged 0.4–3.8, and were found to be dependent on the Reynolds number. Uncertainty in the estimates of the drag coefficient due to the measured projected area and measured wave attenuation was quantified and found to have average combined standard deviations of 0.58 and 0.56 for random and regular waves, respectively. Two previous reduced-scale studies of wave attenuation by mangroves compared well with the present study when their Reynolds numbers were re-scaled by λ3/2 where λ is the prototype-to-model geometric scale ratio. Using the combined data sets, an equation is proposed to estimate the drag coefficient for a Rhizophora mangrove forest: CD = 0.6 + 3e04/ReDBH with an uncertainty of 0.69 over the range 5e03 < ReDBH < 1.9e05, where ReDBH is based on the tree diameter at breast height. These results may improve engineering guidance for the use of mangroves and other emergent vegetation in coastal wave attenuation.

Wave Attenuation Analysis for Artificial Coral Reefs Using a Physical Modelling Approach

Kim, T.; Kwon, Y.; Hong, S.; Kim, J.; Kwon, S., and Lee, J., 2021. Wave attenuation analysis for artificial coral reefs using a physical modelling approach. In: Lee, J.L.; Suh, K.-S.; Lee, B.; Shin, S., and Lee, J. (eds.), Crisis and Integrated Management for Coastal and Marine Safety. Journal of Coastal Research, Special Issue No. 114, pp. 529–533. Coconut Creek (Florida), ISSN 0749-0208. Due to global warming and coastal development, the damages in coastal such as sea level rise and changes in coastline caused by scour and erosion are increasing, Various types of breakwaters are applied to prevent the coastal erosion, however, the problem such as scour around the structure is exacerbated, therefore a fundamental solution is needed. In this study, two-dimensional laboratory experiments were conducted in the wave flume to investigate the wave transmission phenomena of artificial coral reef. Experiments are conducted by analyzing wave transmission dependents on ratio of stem length to water depth(ls/h), wave height and wave period. Wave attenuation appeared to be most dependent on stem length to water depth(ls/h) and structure width. As incident wave height increased and the wave period decreased, the wave attenuation also increased, however, it was insignificant compared to the influence of the inundation rate. Wave attenuation increases dependents on incident wave height and period, however showed insignificant effect compared to submergence rate. In addition wave attenuation rate also affects wave energy reduction in order of ls/h > H0/h > h/L0. Based on results, we analyzed correlation between KC and CD of artificial coral reef to suggest the equation on empirical relationship. As a result, it is judged that estimating wave transmission rate and drag coefficient of artificial coral reef dependents on sectional specification can be utilized properly in working design and application.

Hull form optimization based on calm-water wave drag with or without generating bulbous bow

In the ship design process, hull form design and optimization play an important role. In the research and engineering fields of hull form optimization, the resistance performance is most noticed among all the hydrodynamic performances. In this paper, we focus on the calm-water drag optimization for a high-speed slender hull without a bulbous bow, and the generation of the bulbous bow is introduced in detail, such as the hull form deformation methods and the knot-insertion procedure for the NURBS surface. Two optimization cases are given. In case 1, only shifting method is used, while in case 2, shifting and Radial Basis Function (RBF) methods are both used. During the hydrodynamic evaluation of the new sample hulls, the potential flow solver is applied, and the verification and validation procedure is done at first. The genetic algorithm is used to obtain two optimal ships with minimum wave drag. Optimization results show that even when we do not generate a bulbous bow, the optimal hull will have a relatively big decrease of the wave drag coefficient. However, favorable interference of the wave systems conducted by the generated bulbous bow can help reduce the wave drag coefficient to a greater extent, showing that the bulbous bow generation method proposed in this paper has potentials in the resistance optimization of the medium- or high-speed ship hulls.

Wave transmission and drag coefficients through dense cylinder arrays: Implications for designing structures for mangrove restoration

Mangrove vegetation constitutes a natural coastal defence against waves and erosion. Despite their protective role, mangrove ecosystems have experienced continuous degradation over the last decades due to human causes. At retreating mangrove coastlines, bamboo structures are built to create new habitat for mangrove colonization. Existing structures have experienced mixed rates of success due to the lack of a scientific basis in their design. Optimizing future structure designs requires investigating the effect of the bamboo poles on waves. We consequently conducted laboratory experiments to measure wave transformation, hydrodynamic forces, and flow velocities inside cylinder arrays, mimicking bamboo poles, with varying cylinder configurations and orientations. The experiments provided relationships for wave transmission, wave reflection, and the drag coefficients for configurations with volumetric porosities between n = 0.64 − 0.9. Configurations with a small lateral spacing (causing higher blockage) and a relatively longer streamwise spacing (causing less sheltering) exhibit larger forces and dissipation per element. Such arrangements enable optimizing wave dissipation at locations where the wave direction has low variability over the year. Placing the poles horizontally instead of vertically increases the forces and wave dissipation per element in relatively deeper water. Based on the experiments, we developed a conceptual analytical model that predicts wave reflection and dissipation through cylinder arrays, including blockage and sheltering. The model can reproduce the influence of cylinder arrangement on wave transformation, and it suggests that accurate predictions of sheltering and wave reflection are important to find optimal designs. Overall, these results provide useful insights on how to model and optimize the design of structures for mangrove restoration.

Near-sonic pure steam flow with real-gas effects and non-equilibrium and homogeneous condensation around thin airfoils

A small-disturbance asymptotic model is derived to describe the complex nature of a pure water vapour flow with non-equilibrium and homogeneous condensation around a thin airfoil operating at transonic speed and small angle of attack. The van der Waals equation of state provides real-gas relationships among the thermodynamic properties of water vapour. Classical nucleation and droplet growth theory is used to model the condensation process. The similarity parameters which determine the flow and condensation physics are identified. The flow may be described by a nonlinear and non-homogeneous partial differential equation coupled with a set of four ordinary differential equations to model the condensation process. The model problem is used to study the effects of independent variation of the upstream flow and thermodynamic conditions, airfoil geometry and angle of attack on the pressure and condensate mass fraction distributions along the airfoil surfaces and the consequent effect on the wave drag and lift coefficients. Increasing the upstream temperature at fixed values of upstream supersaturation ratio and Mach number results in increased condensation and higher wave drag coefficient. Increasing the upstream supersaturation ratio at fixed values of upstream temperature and Mach number also results in increased condensation and the wave drag coefficient increases nonlinearly. In addition, the effects of varying airfoil geometry with a fixed thickness ratio and chord on flow properties and condensation region are studied. The computed results confirm the similarity law of Zierep & Lin (Forsch. Ing. Wes. A, vol. 33 (6), 1967, pp. 169–172), relating the onset condensation Mach number to upstream stagnation relative humidity, when an effective specific heat ratio is used. The small-disturbance model is a useful tool to analyse the physics of high-speed condensing steam flows around airfoils operating at high pressures and temperatures.

Wave drag on asymmetric bodies

An asymmetric body with a sharp leading edge and a rounded trailing edge produces a smaller wave disturbance moving forwards than backwards, and this is reflected in the wave drag coefficient. This experimental fact is not captured by Michell’s theory for wave drag (Michell Lond. Edinb. Dubl. Phil. Mag. J. Sci., vol. 45 (272), 1898, pp. 106–123). In this study, we use a tow-tank experiment to investigate the effects of asymmetry on wave drag, and show that these effects can be replicated by modifying Michell’s theory to include the growth of a symmetry-breaking boundary layer. We show that asymmetry can have either a positive or a negative effect on drag, depending on the depth of motion and the Froude number.

The forces of tsunami waves on a vertical wall and on a structure of finite width

The hydrodynamic forces on coastal structures impacted by tsunami depend on the wave Froude number, the sizes and shapes of the structures and also on whether the structures stand in water or on dry land at the time of the impact. We conducted simulations using shallow-water hydraulic equations and then simulations using full Navier-Stokes equations to determine the wave impact on a vertical wall – i.e., a structure of infinite width – and a structure of finite width. In the case of steep waves on a wet bed, the wave-drag coefficient rose to a peak on impact and then settled to a wave-structure interaction equilibrium. In the case of smooth waves on dry land, the approach to the equilibrium was gradual. The numerical simulations determined the dependence of the wave-drag coefficients on the Froude number and the width-to-depth ratio for waves on wet beds and, separately, for waves on dry land. The simulations also reproduced the wave forces for the range of conditions observed in the laboratory.

Computational Analysis of the Stability of the Brazilian Atmosphere Reentry Satellite (SARA)

This paper presents a computational analysis of the stability of the small Brazilian space capsule known as SARA (Atmospheric Reentry Satellite). SARA capsule was developed by the Institute of Aeronautics and Space (IAE) in cooperation with Brazilian Space Agency (AEB) to perform experiments in microgravity environment and return to the Earth. Thus, supersonic and hypersonic flows over the SARA capsule are analyzed using data of thermodynamics and transport properties for equilibrium air. Comparisons based in 2D configurations with different attack angle (0°, 10°, 20°, and 30°) and for middle- to low-altitude portion of SARA (60 km, 45 km, 30 km, and 15 km). The emphasis of this paper is to identify, via computational fluid dynamics analysis, the shock wave positions, drag and lift coefficients and pressure center position, for SARA capsule. In this sense, the results shown that the SARA capsule will be stable during its trajectory, for angle of attack range up to 10°, because lower values of drag and lift coefficients were obtained over that limit. Besides, the center of pressure is located behind center of gravity of the capsule, being another important result that promotes the stability of SARA capsule.

Active Subspaces of Airfoil Shape Parameterizations

Design and optimization benefit from understanding the dependence of a quantity of interest (e.g., a design objective or constraint function) on the design variables. A low-dimensional active subspace, when present, identifies important directions in the space of design variables; perturbing a design along the active subspace associated with a particular quantity of interest changes that quantity more, on average, than perturbing the design orthogonally to the active subspace. This low-dimensional structure provides insights that characterize the dependence of quantities of interest on design variables. Airfoil design in a transonic flowfield with a parameterized geometry is a popular test problem for design methodologies. Two particular airfoil shape parameterizations are examined (parametric sections and class-shape transformation), and the active subspaces present in two common design quantities of interest (transonic lift and wave-drag coefficients) are studied under each shape parameterization. The two parameterizations are mathematically related by a simple change of variable and subsequent odd-power polynomial series. The active subspaces enable low-dimensional approximations of lift and drag that relate to physical perturbations of the airfoil. In particular, a two-dimensional approximation of both transonic lift and drag is obtained and interpreted, where 1) each dimension is a particular linear combination of the design variables resulting in intuitive airfoil perturbations, and 2) the linear combination weights are related between lift and drag. It is shown how these approximations along with similarities between the lift and drag active subspaces inform a multi-objective design problem.

Relationship between turbulent energy dissipation and gas transfer through the air–sea interface

The nonlinear dependence of gas transfer velocity on wind speed typically results in the best fit of observational data; however, gas transfer velocity is dimensionally inconsistent with the nonlinear wind speeds. The objective of the current study was to show that, in the case of wind waves, gas transfer velocity with a consistent dimension should be determined by turbulent viscosity instead of by the viscosity of water when parameterised using the renewal model. Turbulent kinetic energy (TKE) near the air–sea interface is significantly intensified by breaking wind waves. By analyzing various models, we found that the TKE dissipation rate was explicitly linearly related to wind speed and dependent on wave age, with powers ranging from –2.36 to 4.0. Various models show that wave energy dissipation (WED) due to wind wave breaking explicitly increases with the cube of the wind speed and weakly depends on wave age. Assuming a balance between WED and total TKE dissipation in a constant dissipation layer, the depth of this layer was shown to be comparable to the wave height. Using the traditional renewal model with wind wave turbulent viscosity and TKE dissipation rate at the sea surface, we found that the gas transfer velocity was explicitly linearly related to wind speed in a dimensionally consistent manner, and depended simultaneously on the wave age and drag coefficient. These results are consistent with observational data obtained using the eddy correlation method. We emphasise that the linear dependence on wind speed is only valid when the wave age and drag coefficient are fixed; thus, this finding cannot be directly confirmed by currently available observational data due to a lack of wave state information.

Concentrated energy addition for active drag reduction in hypersonic flow regime

Numerical optimization of hypersonic drag reduction technique based on concentrated energy addition is presented in this study. A reduction in wave drag is realized through concentrated energy addition in the hypersonic flowfield upstream of the blunt body. For the exhaustive optimization presented in this study, an in-house high precision inviscid flow solver has been developed. Studies focused on the identification of “optimum energy addition location” have revealed the existence of multiple minimum drag points. The wave drag coefficient is observed to drop from 0.85 to 0.45 when 50 Watts of energy is added to an energy bubble of 1 mm radius located at 74.7 mm upstream of the stagnation point. A direct proportionality has been identified between energy bubble size and wave drag coefficient. Dependence of drag coefficient on the upstream added energy magnitude is also revealed. Of the observed multiple minimum drag points, the energy deposition point (EDP) that offers minimum wave drag just after a sharp drop in drag is proposed as the most optimum energy addition location.

A Numerical Investigation of Cumulus Thermals

Abstract Although the steady, entraining, updraft plume is widely taken as the foundational concept of cumulus convection, past studies show that convection is typically dominated by thermals that are transient, more isotropic in shape, and possess interior vortical circulations. Here, several thousand such thermals are tracked in cloud-resolving simulations of transient growing convective events. Most tracked thermals are small (with radius R < 300 m), ascend at moderate rates (~ 2–4 m s−1), maintain an approximately constant size as they rise, and have brief (4–5 min) lifetimes, although a few are much larger, faster, and/or longer lived. They show slight vertical elongation, but few, if any, would be described as plumes. As convection deepens, thermals originate higher up, are larger, and rise faster, although radius and ascent rate are only weakly correlated among individual thermals. The main force opposing buoyancy is a nonhydrostatic pressure drag, not mixing of momentum. This drag can be expressed in terms of a drag coefficient cd that decreases as convection intensifies: deep convective thermals are less damped, with cd ~ 0.2, while shallow convective thermals are more damped, with cd ~ 0.6. The expected dependence of cd based on theoretical form and wave drag coefficients for a solid sphere is inconsistent with these results, since it predicts the opposite dependence on the Froude number. Thus, a theory for drag on cumulus thermals is not straightforward. Overall, it is argued that thermals are a more realistic prototype for atmospheric deep convection than plumes, at least for the less organized convection types simulated here.

Numerical simulation of free surface flows around shallowly submerged hydrofoil by OpenFOAM

The validation of the Computational Fluid Dynamics (CFD) methodology for numerical simulation of two-dimensional unsteady incompressible viscous flow generated by a shallowly submerged hydrofoil under the free surface is described in this study. The computations are performed using a finite volume discretization incorporated with the interface capturing volume of fluid (VOF) method to solve the fluid equations in motion. The standard k–ε turbulence model is used to capture turbulent flow around the hydrofoil in the free surface zone at different submergence depth ratios. The wave profiles, pressure contours, velocity contours, lift and drag coefficients are compared for the different Froude numbers based on submergence depth to demonstrate the influence of free surface on performance of the hydrofoil. A comparison of numerical and experimental results show that an accurate representation of the free surface flows can be obtained with the present numerical scheme.

Effect of Fineness Ratio on Minimum-Drag Shapes in Hypersonic Flows

The importance of fineness ratio in determining the aerodynamic shapes of minimum-drag, zero-lift axisymmetric bodies in inviscid hypersonic flows is investigated using a shape optimization framework. The framework employs modified Newtonian theory for surface pressure computation, Bezier curves for geometric parameterization and a steepest-descent approach for minimization of the wave-drag coefficient. Studies are performed on axisymmetric bodies of a given length for fineness ratios varying from 1 to 6 and Mach numbers ranging from 6 to 12. It is shown that the optimal axisymmetric body for a given freestream Mach number is blunt-nosed for smaller fineness ratios but becomes sharp-nosed at larger fineness ratios. The fineness ratio at which the optimal body of revolution transitions from blunt-nosed to sharp-nosed is found to be nearly constant at hypersonic speeds and is around 3. Results indicate that the optimal bodies derived from the framework are superior in terms of wave drag to the von Kármán ogive but comparable to the 0.7 power-law body. High-resolution Euler simulations of these optimal forebodies demonstrate that, while the modified Newtonian theory underpredicts the wave drag for lower values of the similarity parameter, the derived configurations remain “near-optimal” for the entire range of the similarity parameter in the hypersonic regime. The study hints at the possibility of multiple near-optimal configurations in inviscid hypersonic flows and affirms the utility of the present framework as a low-fidelity approach for the design of minimum-drag configurations.