Jesse Capecelatro

Abstract Thin structures such as heart valves and aortic dissection flaps interact dynamically with blood flow in human vessels. Their flexibility and capacity for large deformations generate complex, highly transient hemodynamic patterns over the cardiac cycle. Accurately resolving these interactions remains challenging for conventional boundary-fitted fluid–structure interaction approaches. We present an immersed boundary method for simulating thin structures in incompressible flow on unstructured grids. The method couples a stabilized finite element fluid solver with a nonlinear, rotation-free shell formulation through a direct forcing immersed boundary approach. The framework supports both weak (explicit) and strong (implicit) time-coupling strategies, enabling stable simulations over a wide range of solid-to-fluid density ratios. Hydrodynamic forces acting on thin structures are computed from fluid solutions sampled on both sides of the structure, allowing accurate force reconstruction for zero-thickness shells. To our knowledge, this is the first immersed boundary formulation that couples an unstructured finite element fluid solver with a two-dimensional, rotation-free shell model to simulate interactions between thin structures and incompressible flow. Fluid–structure coupling is achieved using predefined finite element shape functions, which provide consistent projection between Eulerian and Lagrangian fields without additional interpolation procedures. The framework is validated using three-dimensional benchmark problems involving thin structures. Then, valve-like model is used to compare strong and weak coupling strategies. Finally, the method is applied to an idealized type-B aortic dissection model. The proposed approach is implemented within the open-source software CRIMSON, a finite element platform for cardiovascular simulation.

Continuous ambulatory monitoring of peripheral vascular perfusion could enable earlier detection of vascular dysfunction in individuals with diabetes mellitus and more timely management of cardiovascular disease. Clinical imaging modalities provide high-fidelity vascular information but are impractical for ambulatory use, whereas most wearable devices are limited to single-modality sensing and do not provide imaging. Electrical bioimpedance has the potential to bridge this gap by enabling rapid spatial and temporal imaging while remaining sensitive to hemodynamic changes. Here, we introduce a wearable ring with 8 electrodes and 32-channel bioimpedance sensing for finger blood flow imaging. In 96 healthy participants measured at rest and during autonomic maneuvers, we resolve conductivity images in the digital arteries associated with pulsatile blood flow and train neural network models for continuous cuffless blood pressure waveform estimation. We demonstrate the feasibility of bioimpedance imaging in a ring form factor, supporting its potential for ambulatory cuffless hemodynamic monitoring.

Continuous ambulatory monitoring of peripheral vascular perfusion could enable earlier detection of vascular dysfunction in individuals with diabetes mellitus and more timely management of cardiovascular disease. Clinical imaging modalities provide high-fidelity vascular information but are impractical for ambulatory use, whereas most wearable devices are limited to single-modality sensing and do not provide imaging. Electrical bioimpedance has the potential to bridge this gap by enabling rapid spatial and temporal imaging while remaining sensitive to hemodynamic changes. Here, we introduce a wearable ring with 8 electrodes and 32-channel bioimpedance sensing for finger blood flow imaging. In 96 healthy participants measured at rest and during autonomic maneuvers, we resolve conductivity images in the digital arteries associated with pulsatile blood flow and train neural network models for continuous cuffless blood pressure waveform estimation. We demonstrate the feasibility of bioimpedance imaging in a ring form factor, supporting its potential for ambulatory cuffless hemodynamic monitoring.

Continuous ambulatory monitoring of peripheral vascular perfusion could enable earlier detection of vascular dysfunction in individuals with diabetes mellitus and more timely management of cardiovascular disease. Clinical imaging modalities provide high-fidelity vascular information but are impractical for ambulatory use, whereas most wearable devices are limited to single-modality sensing and do not provide imaging. Electrical bioimpedance has the potential to bridge this gap by enabling rapid spatial and temporal imaging while remaining sensitive to hemodynamic changes. Here, we introduce a wearable ring with 8 electrodes and 32-channel bioimpedance sensing for finger blood flow imaging. In 96 healthy participants measured at rest and during autonomic maneuvers, we resolve conductivity images in the digital arteries associated with pulsatile blood flow and train neural network models for continuous cuffless blood pressure waveform estimation. We demonstrate the feasibility of bioimpedance imaging in a ring form factor, supporting its potential for ambulatory cuffless hemodynamic monitoring.

Digital twins-virtual representations dynamically linked to physical systems-have the potential to transform biomedical engineering by enabling real-time prediction, optimization, and personalization in health and disease. In biofluids, digital twins offer a framework for integrating physics-based models with data from clinical imaging, sensors, and physiological measurements to support diagnostics, therapeutic planning, and device design. This article reviews modeling approaches used in the construction of digital twins for biofluid applications. We survey high-fidelity numerical methods alongside emerging machine learning techniques, highlighting their respective strengths and limitations. Key requirements for digital twins are discussed, emphasizing the bidirectional interaction between physical and virtual assets and the importance of selecting modeling strategies tailored to specific biomedical contexts. While notable progress has been made over the past decade, significant challenges remain, particularly in integrating multiphysics models with data-driven methods and in establishing standardized protocols for data acquisition, interoperability, and sharing.

Dust and sand ingestion in gas turbine engines leads to particle deposition and wear, degrading performance and increasing maintenance costs. In this work, we use direct numerical simulation of turbulent channel flow to investigate the role of near-wall turbulence in shaping particle deposition patterns. Particle dynamics are resolved via a one-way coupled Euler–Lagrange framework incorporating drag, lift, Brownian dynamics, and a soft-sphere collision model with adhesive contact. Thermal effects are approximated by varying the particle adhesion number. We find that lower adhesion numbers produce more heterogeneous deposits by allowing particles to move along the wall, forming streak-like patterns that align with near-wall turbulence structures. Spanwise radial distribution functions and spatial velocity correlations reveal a direct coupling between turbulence scales and particle clustering. These results highlight the importance of accurate models for particle-turbulence and particle-wall interactions in predicting deposition morphology in high-temperature turbulent flows.

During propulsive landings on the Moon, Mars, or other terrestrial bodies, high-speed thruster plumes erode the regolith surface, generating ejecta that lift off and pose hazards to the vehicle and surroundings. The volume and shape of craters formed during plume-surface interactions (PSI) depend on thruster characteristics, ambient conditions, and regolith properties. We review existing analytical, empirical, and data-driven models of plume-induced granular erosion, highlighting the need for models that accurately capture PSI across a broader range of conditions. In this work, low-pressure, high-speed data from NASA’s Physics-Focused Ground Test (PFGT) campaign were used to characterize crater evolution and morphology across varying nozzle mass flow rates, nozzle heights, background pressures, and regolith simulants. Half-space experiments employed a splitter plate aligned with the nozzle axis, enabling high-speed recordings of crater formation. Crater profiles were extracted using an in-house image processing tool, augmented with an open-source machine learning algorithm. Sensitivity analysis of crater dynamics was conducted to classify cratering behavior based on nozzle, ambient, and particle properties. An intuitive regime map for crater shapes is presented, using nozzle pressure ratio, nozzle height, and a non-dimensional erosion parameter, revealing five distinct cratering regimes. Finally, volumetric erosion rates are compared with existing physics-based models, and a correction is proposed that accounts for nozzle height and reduced ambient pressure.

Large-eddy simulations are performed to quantify the influence of nozzle geometry on combustion efficiency, local mixing, and blowout resistance in non-assist methane flares. Five canonical nozzle shapes are evaluated under relevant industrial flare conditions, including a circle, low-aspect-ratio ellipse, high-aspect-ratio ellipse, diamond, and square. Cornered geometries are shown to enhance near-field recirculation, promote mixing, and sustain flame attachment, resulting in up to a 5 % improvement in combustion efficiency compared with streamlined nozzles. Square nozzles perform best, irrespective of wind direction, and maintain combustion efficiency above 96.5 % even at the highest tested crosswind velocities, while streamlined designs exhibit early flame lift-off, reduced recirculation, and efficiency losses. Sharp-edged nozzles also accelerate scalar homogenization and buffer flames against crosswind-induced strain, significantly improving blowout resistance. Despite the widespread use of circular nozzles in industry, these results highlight a passive geometric modification as a practical route to enhanced flare performance. • Nozzle corners enhance recirculation and increase flare combustion efficiency. • Square nozzles maintain > 96.5 % efficiency under strong crosswinds. • Cornered shapes delay blowout and outperform circles regardless of wind direction. • Mixing analysis reveals sharp edges accelerate fuel–air homogenization.

Abstract Waste gas flares frequently encounter turbulent crosswinds, which pose significant challenges to maintaining the EPA-mandated 96.5% combustion efficiency for non-assist flares. Strong crosswinds can distort flame shapes, disrupt mixing, challenge emissions measurements due to variable speeds and directions, and ultimately degrade flare efficiency. This study quantifies the impact of crosswind turbulence intensity on non-assist flare combustion efficiency using large-eddy simulations coupled with a flamelet progress variable approach. Results show that while jet-induced turbulence enhances mixing and improves combustion efficiency, turbulence from crossflows increases local strain rates and consistently reduces efficiency. Combustion efficiency drops by up to 10% at turbulence intensities approaching 20%. A new correlation for combustion efficiency, obtained using symbolic regression, captures both experimental and simulation data well across natural gas flare flow rates of 2–4 m/s and wind speeds of 0–10 m/s. Incorporating a power-law dependence on turbulence intensity significantly reduces data scatter.

This study employs three-dimensional particle-resolved simulations of planar shocks passing through a suspension of stationary solid particles to study wake-induced gas-phase velocity fluctuations, termed pseudo-turbulence. Strong coupling through interphase momentum and energy exchange generates unsteady wakes and shocklets in the interstitial space between particles. A Helmholtz decomposition of the velocity field shows that the majority of pseudo-turbulence is contained in the solenoidal component from particle wakes, whereas the dilatational component corresponds to the downstream edge of the particle curtain where the flow chokes. One-dimensional phase-averaged statistics of pseudo-turbulent kinetic energy (PTKE) are quantified at various stages of flow development. Reduction in PTKE is observed with increasing shock Mach number due to decreased production, consistent with single-phase compressible turbulence. The anisotropy in Reynolds stresses is found to be relatively constant through the curtain and consistent over all the conditions simulated. Analysis of the budget of PTKE shows that the majority of turbulence is produced through drag and balanced by viscous dissipation. The energy spectra of the streamwise gas-phase velocity fluctuations reveal an inertial subrange that begins at the mean interparticle spacing and decays with a power law of $-5/3$ and steepens to $-3$ at scales much smaller than the particle diameter. A two-equation model is proposed for PTKE and its dissipation. The model is implemented within a hyperbolic Eulerian-based two-fluid model and shows excellent agreement with the particle-resolved simulations.

Flaring serves as an important safety and emissions compliance tool in industries such as oil and gas production, refineries, and landfills. Nonassisted, low-flow (≤100 thousand cubic feet per day (MSCFD)), utility (pipe) flares are widely used in practice, yet there are limited studies of real-world conditions. Additionally, while shrouds (windshields) are commonly used to mitigate wind effects, their impact on flare performance is previously undocumented. This study introduces a novel outdoor testing facility designed to evaluate low-flow flares and quantitatively assess their performance with and without shrouds. Experiments were conducted at flare-gas flow rates of 5 to 75 MSCFD using natural gas and an 80% natural gas/20% propane blend (by volume) under crosswind speeds from 0 to over 35 miles per hour (MPH). Combustion efficiency (CE) and methane destruction removal efficiency (DRECH4) were determined for all operating conditions. While CE for a baseline utility flare (3-inch diameter pipe equipped with a pilot ignition system) was over 96.5% for crosswinds below 10 MPH, the CE decreased rapidly for crosswinds above 10 MPH, with CE <70% for crosswinds above 30 MPH. The utility flare results were compared with results of prior wind-tunnel studies and prior proposed scaling relationships and incorporated into machine learning (ML) models. The scaling relationships show poor correlation with the body of data, but the ML models yielded good agreement (R2 = 0.84) when crosswind turbulence intensity was incorporated as an input parameter. The current work investigated retrofitting a utility flare with different shroud designs, which increased CE ≥96.5% for all conditions, demonstrating the effectiveness of shrouds as practical and cost-effective strategies to improve utility flare performance. The results showed low sensitivity to different shroud designs.Implications: The U.S. Environmental Protection Agency (EPA), industry and other monitoring organizations commonly assume flares operate at 98% destruction efficiency; however, recent aerial surveys have revealed efficiencies as low as 91.1%, resulting in up to five times more methane emissions than expected. Low-flow (≤100 MSCFD) utility flares, widely deployed at oil and gas production sites, have limited performance data under real world conditions. This study addresses that gap by providing new experimental data on low-flow utility flares, identifying a new parameter important for predicting flare efficiency and demonstrating a practical solution for significantly reducing emissions.

We present results from direct numerical simulations of turbulent channel flow laden with adhesive (viscoelastic) particles. Particles demonstrate higher adhesion strengths at elevated temperatures, an effect we probe by varying the adhesion number. Using spanwise radial distribution functions, we show that particle heterogeneity near and on the wall is promoted by turbulence. Furthermore, low-adhesion, high-inertia particles demonstrate spanwise creep along the wall, leading to elongated streamwise deposits. Abrasive wear profiles highlight the consequences of heterogeneity, with local wear exceeding ten times the mean.

• An Euler–Lagrange framework for CO 2 adsorption is presented. • Effect of unresolved particle-induced wakes (pseudo-turbulence) is accounted for. • The optimal filter width during two-way coupling is proportional to the average interparticle spacing. • The proposed framework yields good prediction of CO 2 adsorption in packed bed reactors. Numerical simulations of solid sorbent adsorption must account for processes that span a range of scales involving multiple phases. The Euler–Lagrange (EL) approach is particularly attractive because it resolves particle-scale chemistry without the need to fully resolve particle surfaces. Two key challenges in this approach are selecting an appropriate filter size used during two-way coupling and accounting for unresolved particle-induced turbulence, termed pseudo-turbulence, which enhances scalar mixing. This work addresses both issues by presenting an EL framework for simulating CO 2 adsorption with a focus on properly handling two-way coupling and accounting for the often neglected pseudo-turbulence terms. It is found that an optimal filter size exists when projecting particle data to the grid that is equal to 3.5 times the inter-particle spacing. The influence of unresolved particle-induced wakes on mass, momentum, and heat transport is accounted for using recently proposed models for pseudo-turbulence. Validation against particle-resolved direct numerical simulations confirms the importance of accounting for pseudo-turbulence. The framework is then applied to CO 2 adsorption in a packed bed reactor. CO 2 breakthrough curves are found to be in excellent agreement with experimental data and temperature trends are accurately predicted.

From the crewed Apollo missions to the recent Chinese Chang’e landings, the interaction between spacecraft exhaust plumes and lunar soil produces dusty clouds with high-speed particle ejection. Despite varying landing sites, remarkably stable streak patterns were observed, raising questions about their origin. We solved this puzzle by showing that these patterns were driven by Görtler instability from the curved compressed shear layer of the supersonic but surprisingly laminar jet. This instability creates vortical structures that entrain and eject particles. The number of streaks exhibits an interesting scaling with the jet pressure ratio, which can be modeled with linear instability theory and shows excellent agreement with scaled-down experiments, simulations, and actual observations in landing videos. Our findings provide a fluid physics explanation of extraterrestrial landings, highlighting the role of particle-laden flows and paving the way for future missions to optimize landing strategies and mitigate dust cloud effects on equipment and visibility. Despite observed routinely from spacecrafts landing on, e.g., lunar soil, the origin of radial streak patterns has been unclear up to now. Here, the authors report an experimental study of such instabilities in the coupled dynamics of rocket plumes and sand surfaces.

The interaction of a planar shock wave with a suspension of bidisperse particles is examined using high-fidelity simulations over a range of shock Mach numbers ([Formula: see text]), particle volume fractions ([Formula: see text]), and particle diameter size ratios ([Formula: see text]) between large ([Formula: see text]) and small ([Formula: see text]) particles. The study aims to quantify the effects of these parameters on size segregation and particle curtain spread. A hybrid numerical framework is developed, combining an immersed boundary method for large particles with Lagrangian particle tracking of small particles. The gas-phase equations are discretized using a high-order, energy-stable finite difference scheme with localized shock capturing. Contact between particles is handled using a soft-sphere collision model with an efficient neighbor detection algorithm to capture collisions between particles of varying sizes. Results indicate that size segregation increases with increasing [Formula: see text] and decreasing [Formula: see text]. Additionally, segregation exhibits a weak proportionality to the size ratio [Formula: see text] across the range of conditions and time horizons considered. Streamwise profiles of the mean particle volume fraction reveal a pronounced downstream spread of smaller particles compared to larger ones.

We present direct numerical simulations (DNS) of particle deposition in a turbulent channel flow, incorporating a viscoelastic soft-sphere collision model with temperature-dependent van der Waals adhesion. Particle-wall contact is governed by an adhesion number that varies with temperature, enabling exploration of a wide range of deposition behaviors. Deposition is strongly heterogeneous, especially for inertial particles, where rolling and sliding enhance nonuniformity. Spanwise radial distribution functions reveal that deposited particles form streaks with characteristic spacing set by near-wall two-point velocity correlations. A clustering metric confirms that high-inertia, low-adhesion particles deposit in elongated, anisotropic patterns due to spanwise migration driven by velocity fluctuations. Finally, it is shown that this heterogeneity in deposition leads to localized wall wear rates exceeding ten times the mean, with the most severe wear associated with particles that are carried to the wall in clusters by sweep events.

This study employs three-dimensional particle-resolved simulations of planar shocks passing through a suspension of stationary solid particles to study wake-induced gas-phase velocity fluctuations, termed pseudo-turbulence. Strong coupling through interphase momentum and energy exchange generates unsteady wakes and shocklets in the interstitial space between particles. A Helmholtz decomposition of the velocity field shows that the majority of pseudo-turbulence is contained in the solenoidal component from particle wakes, whereas the dilatational component corresponds to the downstream edge of the particle curtain where the flow chokes. One-dimensional phase-averaged statistics of pseudo-turbulent kinetic energy (PTKE) are quantified at various stages of flow development. Reduction in PTKE is observed with increasing shock Mach number due to decreased production, consistent with single-phase compressible turbulence. The anisotropy in Reynolds stresses is found to be relatively constant through the curtain and consistent over all the conditions simulated. Analysis of the budget of PTKE shows that the majority of turbulence is produced through drag and balanced by viscous dissipation. The energy spectra of the streamwise gas-phase velocity fluctuations reveal an inertial subrange that begins at the mean interparticle spacing and decays with a power law of $-5/3$ and steepens to $-3$ at scales much smaller than the particle diameter. A two-equation model is proposed for PTKE and its dissipation. The model is implemented within a hyperbolic Eulerian-based two-fluid model and shows excellent agreement with the particle-resolved simulations.

The impingement of high-speed rocket plumes on regolith surfaces during propulsive landings on the surface of the Moon or planetary bodies erodes a crater and lifts ejecta that can compromise the lander and nearby infrastructure. There is a lack of engineering models that can aid risk prediction, stemming from an incomplete understanding of plume-surface interactions (PSI) and the diverse erosion mechanisms dominating in different regimes. In this work, we use data from recent sub-scale PSI experiments performed at NASA to advance the current understanding of erosion phenomenology and cratering predictions. Experiments involved a supersonic jet impinging on a granular bed, with a splitter plate aligned with the jet axis allowing visual access to the crater cross-section. This work uses the high-speed imagery of the crater as it evolves under the action of the jet. Custom image-analysis methodologies are used to extract the profile of the crater as the granular surface is eroded. For some of the test cases, the crater shape can be approximated using a family of standard Gaussian curves, which enables crater profiles to be defined with minimal parameters. Using the symbolic regression framework PySINDy, a dynamical system is identified that reduces the crater evolution to a set of ordinary differential equations. The dynamical system generated and presented in this work showed a good agreement with the experimental data.

The interaction of a planar shock wave with a suspension of bidisperse particles is examined using high-fidelity simulations over a range of shock Mach numbers ($M_s$), particle diameter size ratios ($D/d$), and particle volume fractions ($\Phi_p$). The study aims to quantify the effects of these variables on size segregation and particle curtain spread. A hybrid numerical framework is developed, integrating an immersed boundary method for large particles with Lagrangian tracking of small particles. The gas-phase equations are discretized using a high-order, energy-stable finite difference scheme with localized shock capturing. Contact between particles is handled using a soft-sphere collision model with an efficient neighbor detection algorithm to capture collisions between particles of varying sizes. Results indicate that size segregation intensifies with increasing $M_s$ and $D/d$ and decreasing $\Phi_p$. Streamwise profiles of the mean particle volume fraction reveal a pronounced downstream spread of smaller particles compared to larger ones. These results are being used to inform closure for coarse-grained two-fluid models.

Experiments and numerical simulations of inertial particles in underexpanded jets are performed. The structure of the jet is controlled by varying the nozzle pressure ratio, while the influence of particles on emerging shocks and rarefaction patterns is controlled by varying the particle size and mass loading. Ultra-high-speed schlieren and Lagrangian particle tracking are used to experimentally determine the two-phase flow quantities. Three-dimensional simulations are performed using a high-order, low-dissipative discretization of the gas phase while particles are tracked individually in a Lagrangian manner. A simple two-way coupling strategy is proposed to handle interphase exchange in the vicinity of shocks. Velocity statistics of each phase are reported for a wide range of pressure ratios, particle sizes and volume fractions. An upstream shift of the Mach disk in the presence of particles reveals significant two-way coupling even at low mass loading. A semi-analytic model that predicts the extent of the Mach disk shift is presented based on a one-dimensional Fanno flow that takes into account volume displacement by particles and interphase exchange due to drag and heat transfer. The per cent shift in Mach disk is found to scale with the mass loading, nozzle pressure ratio and interphase slip velocity and inversely with the particle diameter.

Kinematics is the study of motion without reference to the forces or stresses that produce the motion. In fluid mechanics, the Eulerian description of fluid motion is most common. Here, the fluid velocity field is considered in a fixed region of space through which the fluid moves so there are as many as four independent variables – three spatial coordinates and time. In the Eulerian formulation, fluid acceleration is not determined for individual fluid particles. Instead it is determined as a function of all four independent variables and therefore involves both temporal and spatial differentiation of the fluid velocity field . Streamlines, path lines, and streak lines may be used to describe flow field kinematics within the region of interest. Strain-rate and rotation tensors describe the deformation and rotation of infinitesimal fluid particles. For arbitrary but finite regions of space, commonly called control volumes, time derivatives of volume integrals must include the possibility of fluid and/or volume motion through use of the Reynolds Transport Theorem .

Turbulence is an enigmatic state of fluid flow that includes unpredictable fluctuations even when the flow's boundary conditions are steady and smooth. These fluctuations typically occur over a wide range of length and time scales. For analysis, a turbulent flow is decomposed into average (or mean) and fluctuating fields. The field equations governing the average flow field contain derivatives of the correlation tensor of the fluctuating flow field, and therefore do not represent a closed system of equations. However, when the turbulence is homogeneous and isotropic some relationships between the average and fluctuating flow fields can be deduced. In particular, the kinetic energy that sustains the turbulence is extracted from the average flow and is eventually dissipated by the viscous motions of the smallest turbulent fluctuations. Spectral measurements from a variety of turbulent flows suggest this process is universal. Scaling laws for turbulent shear flows that develop near or away from solid boundaries can be deduced from dimensional analysis, similarity analyses, and geometric considerations. Turbulent mean-flow predictions commonly rely on approximate models that close the system of average-flow equations. Turbulence may be modified in stratified media like the earth's ocean and atmosphere.

Only stable steady solutions to the equations of fluid motion are observed in experiments. Unstable solutions may exist briefly but they will spontaneously develop oscillations in space or time that grow and eventually change the character of the flow. Temporal instability analysis presumes an undisturbed background flow into which a disturbance having infinitesimal amplitude is introduced. The equations governing the disturbance are linear and commonly lead to exponential solutions. If the disturbance amplitude decreases, remains the same, or grows in time after its introduction, the flow is said to be stable, neutrally stable, or unstable, respectively. Common flow instabilities may take the form of traveling waves or spatially stationary oscillations in time or space. Depending on the flow's geometry , instability may be induced by the effects of fluid inertia, gravity, rotation, thermal expansion, differential diffusion, fluid viscosity , etc. Once a flow becomes unstable, the instability may grow to be become a new steady-state solution, or it may trigger additional instabilities that also grow. When growing instabilities reach levels where nonlinear interactions occur between them, the flow may be chaotic or even turbulent.

Laminar boundary-layer flows occur when a moving viscous fluid comes in contact with a solid surface and a layer of rotational fluid, the boundary layer, forms in response to the action of viscosity and the no-slip boundary condition on the surface. When the surface is flat or mildly curved and the boundary layer that develops on it is thin and remains adjacent to it, the flow within this layer may be determined by simplifying the Navier-Stokes equations to account for the flow's geometry and then solving the simplified equations. Unfortunately, when the pressure increases in the downstream direction or when the surface is highly curved, the boundary layer may leave the surface, a phenomenon known as separation, and the simplified form of the Navier-Stokes equations no longer applies. In addition, at sufficiently high Reynolds number , boundary-layer flows may spontaneously become unsteady and then transition to turbulence. In combination, these phenomena provide explanations for the fluid dynamic forces felt as fluid moves past a cylinder or sphere at different Reynolds numbers .

Fluids, materials that deform continuously under an applied shear stress , are omnipresent in the world around us, and beyond. Fluid mechanics is the branch of science concerned with stationary and moving fluids. Here fluids are treated as being continuous even though they are composed of discrete molecules. At the macroscopic level, the molecular character of fluids is manifested as diffusive transport of species, heat, and momentum. With the continuum approximation, dependent flow-field variables (velocity, pressure, density, temperature, etc.) are presumed well defined at every point in space and classical equilibrium thermodynamics is presumed valid. In static situations, gravity and thermodynamic gradients determine the stability of the fluid configuration to small perturbations. When fluids move, they obey Newton's second law but there is no restriction on the system of units used to describe the relevant forces and accelerations. This fact and the requirement for dimensional homogeneity in physically meaningful equations allows potentially useful scaling laws to be developed from considerations of the dimensions of relevant parameters, fluid properties, and fundamental constants.

The dependent field variables used to characterize moving and stationary fluids are scalars, vectors, and second-order tensors. In three spatial dimensions, scalars (like density) can be described by a single number; vectors (like velocity) can be described by three components, one for each orthogonal spatial direction; and second-order tensors (like stress) can be described by nine components, one for each pair of spatial directions. Although the meanings of vectors and second-order tensors that represent physical quantities are independent of the orientation of the coordinate system in which they are resolved, their components do change when coordinate axes are rotated. The equations for fluid motion include a variety of algebraic and differential relationships involving scalars, vectors, and second-order tensors. Index notation is a convenient means for specifying these relationships and operations, especially when the gradient operator (vector derivative) is involved. In addition, integrals of flow-field quantities along curves, on surfaces, and within volumes in two and three dimensions must follow certain relationships.

The laws governing fluid motion are based on conservation of mass , momentum, and energy. For the Eulerian description of fluid motion, these three conservation laws are coupled nonlinear partial differential equations . However, to produce a potentially solvable set of equations, a constitutive relationship must be specified. For many commonly encountered fluids, the simplest possible Newtonian viscosity law – a linear relationship between the stress and strain-rate tensors involving only two material constants – is appropriate. When supplemented by two thermodynamic relationships , such as caloric and thermal equations of state , the number of equations matches the number of unknown dependent field quantities. Thus, with the specification of appropriate boundary conditions, the overall system of equations can in principle be solved even in noninertial coordinate systems. When the equations of fluid motion are cast in dimensionless form , the dimensionless parameters (or numbers) commonly used to specify fluid flow conditions appear as coefficients in the equations. Although analytical solutions to the full set of equations are uncommon, the equations of fluid motion can be simplified, and are easier to solve, under certain circumstances. Examples of such principles, equations, boundary conditions, and dimensionless parameters are provided in this chapter.

Aerodynamics is the branch of fluid mechanics that deals with the fluid dynamic forces and moments that act on moving objects. Lift and drag, the aerodynamic force components perpendicular and parallel to the oncoming fluid velocity, are both the result of viscous effects within a fluid flow. For two-dimensional (infinite span) airfoils at low or moderate angles of attack (up to ∼ 15 ∘ ), lift may be predicted by ideal flow analysis when the effect of viscosity is represented by choosing the foil's circulation so that the aft flow-separation point occurs at the foil's trailing edge . For finite-span wings, trailing vortices aligned with the oncoming flow develop downstream of each wing tip and induce a downward vertical velocity at the wing. This induced velocity locally changes the angle of attack along the wing in a manner that rotates the lift force backward to produce a drag force known as induced drag. Fish, birds, and sailboats use aerodynamic lift forces for propulsion and maneuvering.

When a constant-density fluid flows without rotation and pressure is measured with respect to its local hydrostatic value, the field equation for fluid motion is linear and pressure can be determined from a nonlinear algebraic (Bernoulli) equation. Under these ideal-flow conditions, a single scalar stream function or velocity potential may be used to describe the flow. Ideal flows around corners, near walls, and past simple obstacles may be constructed by superimposing fundamental solutions. In two dimensions these functions are naturally combined into a complex potential that satisfies the Cauchy-Riemann equations for a complex analytic function. The complex potential allows two important ideal-flow theorems to be proved, and it allows complicated flows to be analyzed via complex-variable mapping techniques. For axisymmetric and three-dimensional ideal flows, the fundamental equations are still linear but the extension to complex functions is lost. In particular, unsteady three-dimensional potential flow can be used to determine the apparent (or added) mass of a fully submerged, arbitrarily moving sphere.

The vorticity field – defined as the curl of the velocity field , and twice the rotation rate of fluid particles – is a fundamental quantity in fluid mechanics. Similar to how streamlines were obtained from the velocity field , vortex lines may be determined from a tangency condition of the vorticity vector. However, vortex lines have several special properties and their presence or absence within a region of interest may allow certain simplifications of the field equations for fluid motion. In particular, vortex lines are carried by the flow and cannot end within the fluid, which constrains their possible topology. Vorticity is typically present at solid boundaries and may diffuse into the flow via the action of viscosity . Vorticity may be generated within a flow wherever there is an unbalanced torque on fluid elements, such as when pressure and density gradients are misaligned. The characteristics and geometry of a thin vortex allow the velocity it induces at a distant location to be determined. Thus, multiple vortex lines that are free to move within a fluid may interact with each other. In a rotating coordinate frame, the observed vorticity depends or the frame's rotation rate . This chapter introduces basic concepts and phenomena associated with fluid vorticity and derives useful theorems and transport equations in inertial and rotating frames of reference.

Viscous flows occur when the effects of fluid viscosity are balanced by those arising from fluid inertia, body forces, and/or pressure gradients. In such flows, scaling analyses do not allow a priori neglect of any terms in the equations of fluid motion. However, under certain ideal geometrical circumstances involving locally parallel walls that confine the flow, relatively simple steady and unsteady exact solutions to the Navier-Stokes equations are possible because the nonlinear advective acceleration is identically zero. Interestingly, the character of these exact solutions persists when the flow's geometry deviates mildly from ideal, a fact exploited in lubrication theory. When the flow's boundary and initial conditions do not impose a length or time scale, exact solutions may sometimes be determined in terms of a special combination of two independent variables known as a similarity variable. At sufficiently low Reynolds numbers, the influence of fluid inertia may be neglected (the creeping flow approximation) and this allows the low-Reynolds number viscous flow past a sphere to be determined.

A flow is considered compressible when changes in fluid momentum produce important variations in fluid pressure and density, and the fluid's thermodynamic characteristics play a direct role in the flow's development. When the pressure variations are small enough, linear acoustic theory may apply. However, larger finite-amplitude pressure disturbances produce nonlinear effects . Compressible flows in ducts and nozzles may reach limiting mass-flow-rate values that cannot be exceeded even when the downstream pressure is decreased. Here friction and heat addition or extraction may have unexpected consequences. Supersonic flows (Mach number >1) may also contain shock waves that induce nearly discontinuous changes in the flow's state. In supersonic flow, downstream pressure disturbances cannot propagate upstream and oblique expansion or compression waves emanate from locations where the flow changes direction. Thus, supersonic flows are often easier to analyze than subsonic flows because the various influences of geometric features of the flow's boundaries need not be assessed simultaneously as would be the case in subsonic flow.

Geophysical fluid dynamics deals with flows of air and water in the atmosphere and ocean. Here fluid velocities are subsonic, the medium is stratified, and the rotation rate of the earth is important. The latter two phenomena suppress vertical motions so horizontal velocities predominate, especially at large scales, even when the flow is turbulent. In fact, the length scales of the motion are often so large that the advective acceleration is small compared to the Coriolis acceleration (small Rossby number), and the fluid's horizontal velocity is perpendicular (not parallel) to the horizontal pressure gradient . Near surfaces where friction is important, the direction of the flow in a boundary layer depends on the distance from the surface. Coriolis effects also cause surface-wave motions to include particle deflections in both horizontal directions, and for surface waves to be trapped near vertical boundaries. Coriolis effects modify internal waves, too. At even larger scales where the curvature of the earth leads to nontrivial variations in the Coriolis frequency, Rossby waves spanning a significant range of latitude may be subject to barotropic and/or baroclinic instabilities .

Waves may occur at fluid interfaces when gravity or surface tension provides a restoring force that pushes a deformed surface back toward its equilibrium position . The general formulation of the surface wave problem is nonlinear, even when the flow is inviscid. An appropriate linearization for small surface slope leads to traveling-wave solutions that are dispersive in deep water and nondispersive in shallow water . In particular, the phase and group speeds of deep-water gravity waves are different and both depend on wavelength with longer waves traveling faster. In shallow water , gravity-wave phase and group speeds are equal and depend on the depth of the water in which they travel. The complexity of the situation increases when the waves are nonlinear, when they occur between fluid layers of differing density, and when they occur on density gradients within a stratified fluid . In the latter case, the phase and group velocities are not even in the same direction.

Experiments and numerical simulations of inertial particles in underexpanded jets are performed. The structure of the jet is controlled by varying the nozzle pressure ratio, while the influence of particles on emerging shocks and rarefaction patterns is controlled by varying the particle size and mass loading. Ultra-high-speed schlieren and Lagrangian particle tracking are used to experimentally determine the two-phase flow quantities. Three-dimensional simulations are performed using a high-order, low dissipative discretization of the gas phase while particles are tracked individually in a Lagrangian manner. A simple two-way coupling strategy is proposed to handle interphase exchange in the vicinity of shocks. Velocity statistics of each phase are reported for a wide range of pressure ratios, particle sizes, and volume fractions. The extent to which particles affect the location of the Mach disk are quantified and compared to previous work from the literature. Furthermore, a semi-analytic model is presented based on a one-dimensional Fanno flow that takes into account volume displacement by particles and interphase exchange due to drag and heat transfer. The percent shift in Mach disk is found to scale with the mass loading, nozzle pressure ratio, interphase slip velocity, and inversely with the particle diameter.