Fluid dynamics

Fluid dynamics studies the motion of liquids and gases, including the forces and transport processes that determine their evolution. Its central variables include velocity, pressure, mass density and temperature, all varying with position and time.

Conservation of mass gives the continuity equation. Combining Newton's second law with a constitutive model for viscous stress gives the Navier-Stokes equations. The relative importance of inertia and viscosity, measured by the Reynolds number, explains why slow flow around a microscopic particle differs so sharply from a turbulent river. Boundary conditions and material properties are essential parts of a fluid model.

A flow containing two or more distinguishable material phases, such as two immiscible liquids or a gas-liquid mixture. Each phase can have its own volume fraction, density and velocity. Cross-sectional or ensemble averaging produces coupled phase balance equations whose mass, momentum and energy exchange terms must be specified consistently. A common pressure alone does not automatically make an averaged model well posed.

Take incompressible phase mass equations for fractions $\alpha ,1-\alpha$ and momentum equations ${\rho }_i[(u_i{)}_t+u_i(u_i{)}_x]+p_x=0$ with a common pressure. At a uniform state with $0<\alpha <1$, elimination of the mass and momentum perturbation amplitudes givesIts roots are the displayed speeds, whose imaginary parts are $\pm s(u_1-u_2)/(1+s^2)$. A nonzero slip velocity therefore gives a Fourier branch growing as $e^{|k||\mathrm{Im}c|t}$. No finite-order Sobolev estimate can control all high wavenumbers, showing ill-posedness of this unregularized closure. At zero slip the roots coincide and this nonreal-root argument no longer settles the degenerate case. Phase-fraction endpoints also lie outside the root formula's derivation.

Here $⟨\cdot ⟩$ means integration over a fixed pipe cross-section divided by its area, and ${\chi }_1$ is the indicator of one immiscible fluid. The volume fraction is $\alpha$; intrinsic density averages over the phase-occupied area rather than over the whole section. The velocity must be density-weighted for its product with mean phase density to represent mass flux exactly.

If there is no interphase mass transfer, integrating ${\partial }_t(\rho {\chi }_1)+\nabla \cdot (\rho v{\chi }_1)=0$ over a section with impermeable outer wall gives ${\partial }_t({\rho }_1\alpha )+{\partial }_x({\rho }_1\alpha u_1)=0$. The second phase uses $1-\alpha$ and its own intrinsic averages. For constant phase density the velocity reduces to the ordinary phase-area average. These definitions apply where the corresponding phase fraction is positive; zero-fraction intrinsic quantities need a separate convention.

For an incompressible phase, this coefficient compares its actual axial momentum flux with the uniform-profile flux at the same intrinsic mean velocity. Cauchy-Schwarz gives $D_1\ge 1$ where the mean is nonzero. A spatially varying profile can have a larger second moment even when its mean speed is unchanged.

With constant $D_1$, expansion of ${\partial }_t(\alpha u_1)+{\partial }_x(\alpha D_1{u}_1^2)$ and use of ${\alpha }_t+(\alpha u_1{)}_x=0$ gives $\alpha (u_1{)}_t+(2D_1-1)\alpha u_1(u_1{)}_x+(D_1-1)u_1^2{\alpha }_x$. Thus a profile correction changes both convection and phase-fraction-gradient coupling in the averaged momentum equation. If $D_1$ varies, an extra $\alpha u_1^2(D_1{)}_x$ remains; a zero mean should be treated through the undivided second moment rather than an undefined coefficient.

The source ${\Gamma }_i$ is the mass gained by phase $i$ per mixture volume and time, through processes such as evaporation or condensation. Internal exchange conserves total mass, so its phase sources sum to zero. In an immiscible material-interface model without phase change, every ${\Gamma }_i$ is zero. The interface may still move: absence of transfer means that mass does not cross it relative to its own motion.

Laminar flow is a regime of fluid flow in which the motion remains organized, with neighboring fluid parcels following smooth, coherent paths rather than the irregular interacting fluctuations characteristic of turbulence. In a straight pipe, the simplest example consists of coaxial layers sliding past one another. This describes the character of the motion, not a requirement that every parcel have the same velocity. Nor does it require steady flow: a smooth profile can evolve in time as viscosity transmits motion from a newly moving wall into a resting fluid. The unsteady wall-driven solution provides an explicit example. ^[ref]

The Reynolds pipe-and-dye experiment makes this distinction visible. A thin injected dye band can remain coherent in laminar flow, whereas irregular motion can fold it and carry dye across the pipe. However, molecular diffusion can spread dye even in laminar flow, and an inlet disturbance can move a coherent band without producing turbulence. A dye band is a streakline, not a measurement of the entire velocity field; in steady flow an ideal passive tracer's streakline coincides with a streamline. Reynolds explicitly distinguished a wandering band from one broken into eddies. ^[ref]

For a circular pipe of internal diameter $D$, define the mean axial velocity $U$ using the volumetric flow rate $Q$ and cross-sectional area $A$:Here $\rho$ is mass density, $\mu$ is dynamic viscosity and $\nu =\mu /\rho$ is kinematic viscosity. This particular Reynolds number uses the diameter and mean velocity, not the radius or centreline velocity. Small Reynolds number generally favors laminar flow, but a single universal critical Reynolds number does not distinguish it from turbulence in all geometries and experiments. Even in a given pipe, inlet conditions and the magnitude of disturbances affect the observed onset. Reynolds demonstrated this dependence by deliberately inserting a wire coil. ^[ref]

The quantitative example above assumes steady flow, incompressible flow, a Newtonian fluid with constant dynamic viscosity, a straight horizontal circular pipe of radius $R=D/2$, stationary walls satisfying the no-slip boundary condition, and fully developed flow. The last assumption means that the axial velocity profile no longer changes downstream. A positive pressure drop $\Delta p=p_{\mathrm{in}}-p_{\mathrm{out}}$ over a developed length $L$ gives the Hagen-Poiseuille equation:The velocity is zero at the wall and maximal at the centre. Averaging over the circular area, whose annuli have area $2\pi rdr$, gives $U=u(0)/2$; averaging along one diameter would give a different answer. These assumptions produce a parabola, rather than every laminar flow having a parabolic profile. ^[ref]

As a stipulated calculation, take $D=0.010\mathrm{m}$, $U=0.10\mathrm{m}{\mathrm{s}}^{-1}$, $\rho =1000\mathrm{kg}{\mathrm{m}}^{-3}$, $\mu =0.0010\mathrm{Pa}\mathrm{s}$ and $L=1.0\mathrm{m}$, and assume the developed laminar flow above. Then $\nu =10^{-6}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, ${\mathrm{Re}}_D=1000$, $u(0)=0.20\mathrm{m}{\mathrm{s}}^{-1}$, $Q=7.85\times 10^{-6}{\mathrm{m}}^3{\mathrm{s}}^{-1}$ (about $0.471\mathrm{L}\mathrm{mi}{\mathrm{n}}^{-1}$), and $\Delta p=32\mu LU/D^2=32\mathrm{Pa}$. These are predictions from chosen parameters, not measurements or a claim that ${\mathrm{Re}}_D=1000$ guarantees laminar flow in every apparatus.

The beauty of this example is that a spatially varying velocity field still gives an exact relation between two accessible measurements, pressure drop and volumetric flow rate. It predicts the hydraulic resistance $\Delta p/Q=8\mu L/(\pi R^4)$. Doubling the radius therefore multiplies $Q$ by sixteen at fixed pressure drop, length and dynamic viscosity, provided the model remains applicable. This makes the laminar flow model useful for sizing narrow tubes or determining dynamic viscosity. It also explains why a persistent narrow dye band can be informative without implying that the fluid is moving as a uniform plug.

In his 1883 paper, Osborne Reynolds investigated how the character of pipe flow depends jointly on velocity, pipe diameter and viscosity. His final dye apparatus drew water from a large glass tank through a horizontal glass tube with a smooth trumpet-shaped entrance. A separate reservoir supplied strongly colored water just upstream of the entrance; a valve below the tank controlled the outflow. The glass tube was immersed in the tank, so its internal dye band could be watched. Reynolds repeated the trials with different tube diameters and water temperatures. ^[ref]

The volumetric flow rate was measured independently of the appearance of the dye. Reynolds timed a one-inch fall of the tank's water level, corresponding to a calibrated discharged volume. In modern notation, if this volume is $V$ and the elapsed time is $\Delta t$, then $Q=V/\Delta t$ and $U=4V/(\pi D^2\Delta t)$. He determined the glass tubes' internal diameters by filling them with water. For a uniform bore, a known filled length $\ell$ and measured water volume $V_{\mathrm{tube}}$ determine $D=\sqrt{4V_{\mathrm{tube}}/(\pi \ell )}$; checking bore uniformity matters because the same volume would only give an average cross-section in a tapered tube. Thermometers at different tank levels checked both the temperature and its uniformity. Reynolds compared the viscosity at different temperatures using Poiseuille's results and checked his own water by separate experiments of that type. Thus the Reynolds number estimate did not come from judging the dye band or from assuming that transition occurred at a preset value. ^[ref]

A practical repetition can instead collect a known outflow volume in a measuring vessel and time its collection, or weigh the collected water and divide its mass by mass density. Measure the internal diameter, rather than the external diameter, and obtain kinematic viscosity at the measured water temperature from independently established properties or a separate viscosity measurement. Record the dye appearance and observation position alongside $Q$, $D$, temperature and the resulting Reynolds number. These are alternative measurement procedures following $Q=V/\Delta t$, not additional historical claims about Reynolds' dye apparatus.

Begin with settled water and a small dye supply, then open the outlet valve in small steps. Reynolds saw a thin straight band at sufficiently low speeds. With a tank that had not fully settled, the band could wander while retaining its coherence. At higher discharge it could abruptly spread into the surrounding water downstream of the entrance. Short illumination by an electric spark resolved curls within what otherwise looked like a colored cloud. This connected the visible breakup with internal irregular motion rather than merely the appearance of an opaque pipe. ^[ref]

The controls are part of the result. Reynolds used a smooth entrance and let the tank rest to reduce inlet disturbances. He found that a heated tank could retain slow circulation as it cooled, so measuring one temperature was insufficient: thermal uniformity and residual motion also mattered. His wire-coil trial broke up the band below the previous onset speed, but he explicitly cautioned that once the dye was mixed he could not tell from that dye alone whether the motion became orderly again farther along the tube. In a repetition, keep the dye supply small enough to act approximately as a passive scalar, avoid a strong injection jet, and record the inlet condition and downstream observation distance. If also testing the Hagen-Poiseuille equation, allow the profile to develop and measure pressure drop between positions within the developed region; do not treat the entrance and outlet losses as part of its uniform pressure gradient. The dye experiment by itself does not establish fully developed flow. ^[ref]

The experiment's usefulness is that it compares a visible change in the motion with independently measured scales. Repeating it with several diameters and temperatures, while controlling disturbances, tests the organizing role of the Reynolds number. Its limitations are equally instructive: dye is a tracer of transport, diffusion still acts in laminar flow, and a local dye observation cannot by itself prove that turbulence persists throughout an arbitrarily long pipe.

Sediment transport is movement of grains by a flowing fluid, through motion close to a bed or within a particle suspension. A flow's shear stress and fluctuating forces compete with grain weight, contacts and friction. The Shields parameter compares a bed-stress scale with the submerged grain-weight scale and helps organize the threshold for movement.

Bedload transport carries grains along or near the bed, while suspended sediment balances settling against upward mixing and entrainment. The transport budget changes the bed itself: mass conservation in the Exner equation relates downstream flux divergence to erosion or deposition. A flow entering an erodible reach can take a finite distance to establish its transported grain population, described by sediment transport saturation. These couplings make a bed an evolving part of the fluid problem rather than a fixed rough wall. Measuring both transported material and bed-height change tests whether the sediment budget closes.

In a reduced sediment-bed model with normalized discharge one, water momentum gives $s=1-u^{-1}+F^2(1-u^2)/2$, while concentration adaptation and bed exchange give $c_x=E(u)-c=-s_t$. Linearizing about $u=c=1,s=0$ yields $\tilde{s}=(1-F^2)\tilde{u}$ and $(1+ik)\tilde{c}=E^{\prime }(1)\tilde{u}$. Elimination proves the displayed dispersion relation.

For $E^{\prime }(1)>0$, the growth rate is $E^{\prime }(1)k^2/[(F^2-1)(1+k^2)]$, positive exactly on the supercritical side $F>1$ for nonzero $k$. The phase speed is $-E^{\prime }(1)/[(F^2-1)(1+k^2)]$, so unstable bedforms migrate upstream and are antidunes. At high wavenumber growth approaches a finite plateau rather than becoming unbounded. This closure selects no finite fastest wavelength; short-scale selection needs additional physics. The critical limit $F=1$ and a vanishing entrainment derivative are degenerate cases outside the displayed conclusion.

Bedload transport is motion of sediment grains along or close to the bed, represented in a depth-averaged model by a solid-volume flux per unit width. The Exner equation balances this flux divergence and changes in porous-bed solid storage against exchange with suspended sediment. Consistent mass units are necessary when adding this equation to a suspended mass-concentration balance.

Keeping sediment acceleration while neglecting water storage and a small gravity-friction residual gives a reduced model for the bed elevation $s$, water speed $u$ and sediment speed $v$. A prescribed water flux $q$ gives $h=q/u$; integrated water momentum gives $s=C-F^2{u}^2/2-q/u$. Around the normalized uniform state $h=u=q=1$, let $(1-v_0{)}^2=\kappa v_0$, $0<v_0<1$. A Fourier mode of wavenumber $k$ has growth-rate real partThus nonuniform modes decay for subcritical Froude number and grow for supercritical Froude number. The spatially constant mode is neutral, and the critical value needs terms neglected in this reduction. In a general uniform state the sign change is at the actual mean-flow Froude number equal to one, rather than at a velocity-scale parameter unrelated to that state.

Suspended sediment is particulate material carried within a fluid rather than solely along the bed. Its concentration balance includes advection, erosion into suspension and deposition from suspension. For water depth $h$ and suspended mass concentration $c$, the conservative balance $(hc{)}_t+(huc{)}_x={\rho }_s(v_E-v_D)$ reduces using water continuity to $h(c_t+u{c}_x)={\rho }_s(v_E-v_D)$.

A dune is a sediment bedform. In the elementary subcritical flow description of river dunes, bed undulations and free-surface undulations are out of phase. This phase relation identifies a hydraulic response; instability additionally requires a positive real part of the relevant dispersion relation. Contrast the in-phase surface of an antidune.

An antidune is a bedform associated with supercritical free-surface flow, in which bed and surface undulations are approximately in phase. In the elementary erodible-bed model with suspended sediment, an unstable branch can migrate upstream. Its growth and migration depend on the hydraulic and sediment-response parameters; the high-wavenumber antidune dispersion calculation makes the required supercritical regime explicit.

In the fast-water limit of an erodible-bed shallow water equations model, let $e=E^{\prime }(1)>0$, $q_1=q^{\prime }(1)>0$ and $\gamma >0$ be the suspended-load relaxation coefficient. Linearization and elimination give the dispersion relation $\lambda =-k^2[q_1+\beta e/(\gamma +ik)]/[3\delta +ik(F^2-1)]$. Its real part tends to the finite constant in the title, and phase velocity tends to $-q_1/(F^2-1)$. Thus the unstable upstream branch requires supercritical flow and enough sediment feedback to overcome friction; subcritical flow has the opposite propagation sign and decay at leading order. The near-critical limit needs a separate balance.

Sediment resuspension transfers deposited bed particles into a particle suspension. It can supply particle buoyancy flux to a particle-laden gravity current if erosion or upward mixing exceeds settling losses. Ambient fluid without particles dilutes concentration without supplying particle mass.

For downward settling velocity magnitude $W_s>0$ and eddy diffusivity ${\kappa }_T=Kz$, the total upward particle flux is $J=-W_s\varphi -Kz{\varphi }_z$. A stationary profile tending to zero at infinity has $J=0$, giving the displayed power law. The exponent compares settling with turbulent mixing. It describes a region above the bed; entrainment from the bed and viscous near-wall effects determine the reference concentration separately.

Sediment transport saturation is relaxation of a transported grain population toward the equilibrium sediment flux supported by local flow conditions. A change in bed shear stress does not immediately give a fully adjusted grain speed and population. Close to equilibrium, a steady model isFor constant $q_{\mathrm{sat}}$ and saturation length $L_{\mathrm{sat}}>0$, this gives $q=q_{\mathrm{sat}}+[q(0)-q_{\mathrm{sat}}]e^{-x/L_{\mathrm{sat}}}$. The spatial lag matters for bedform growth.^[ref]

A published wind tunnel test used a $1\mathrm{m}$ wide, $0.5\mathrm{m}$ high and $4.5\mathrm{m}$ long tunnel. Repeated measurements of bed elevation after stopping the flow supplied an erosion rate; integrating that rate inferred downstream flux by mass conservation. The fitted near-equilibrium relaxation length was distinguished from the earlier grain-ejection stage, so the total development distance was not treated as the same parameter.^[ref]

The equilibrium sediment flux is the transport supported locally by a prescribed bed shear stress. A common idealization is $q_{\mathrm{sat}}={\varphi }_b\chi (\tau -{\tau }_{\mathrm{th}}{)}_{+}^{\gamma }$. Linearization above the sediment entrainment threshold requires perturbations small compared with the positive excess stress.

The saturation length measures downstream relaxation of a transported grain population toward its equilibrium sediment transport flux. The elementary steady relaxation law is $L_{\mathrm{sat}}{q}_x=q_{\mathrm{sat}}-q$; it introduces a phase lag between forcing and actual transport.

The Exner equation is local sediment mass conservation expressed in terms of solid volume: accumulation raises the bed and flux divergence lowers it. Here ${\varphi }_b$ is the solid volume fraction of the packed bed and $q$ is solid volume flux per unit width.

A leading fixed-discharge, level-surface approximation to a shallow erodible-bed flow gives perturbation equations $C_x=4U-C$ and $U_t+3\beta U_x=-4U+C$. The displayed Fourier growth rate has negative real part for every nonzero wave number, independently of bedload parameter $\beta$. Bedload changes the propagation speed but not this damping. The zero-wave-number mode is neutral, so absence of growing modes should not be confused with strict asymptotic decay of every spatially uniform perturbation.

The Shields parameter is $\Theta =\tau /[({\rho }_p-{\rho }_f)gd]$, comparing bed shear stress with a grain-scale submerged weight per area. Its critical value depends on the assumed drag coefficient, static friction, lift, and contact geometry.

The sediment entrainment threshold is the bed shear stress at which resting grains begin moving. An elementary force balance using spherical grains, tangential quadratic drag, and static friction gives ${\Theta }_{\mathrm{th},0}=4{\mu }_s/(3C_D)$ when the drag coefficient is defined relative to the bed shear velocity.

For locally bed-tangent drag force and an upslope angle $a$, a spherical-grain force balance gives ${\tau }_{\mathrm{th}}(a)={\tau }_{\mathrm{th},0}(\cos a+\sin a/{\mu }_s)$. The small-slope term is $\delta {\tau }_{\mathrm{th}}=({\tau }_{\mathrm{th},0}/{\mu }_s){\eta }_x$. A horizontal drag force instead changes the normal contact force and gives a different slope correction.

A bedform is organized relief on a mobile sediment bed, such as a dune or an antidune. Its shape changes the overlying flow, which changes the shear stress and the sediment transport; erosion and deposition then feed back on the shape. The pattern is therefore a coupled fluid-and-grain response rather than simply a frozen pile of material.

For a small sinusoidal bed disturbance, an upstream lead of the forcing relative to bed height can favor growth, while sediment transport saturation introduces a downstream lag. The Exner equation converts the resulting flux divergence into bed-height change. Their combination determines both growth and migration; the direction of migration cannot be inferred from a bed's appearance alone. Recording a bed profile at successive times separates changing amplitude from movement of a crest, while flow measurements test the assumed bed shear response.

Combining the Exner equation, sediment transport saturation, bed shear response, and inclined-bed sediment threshold gives a competition between upstream forcing and downstream relaxation. With $a={\tau }_{0}A$, $b={\tau }_{0}B-{\tau }_{\mathrm{th},0}/{\mu }_s$, $C=\chi \gamma ({\tau }_0-{\tau }_{\mathrm{th},0}{)}^{\gamma -1}$, and positive wavenumber $k$, the growth rate is $\sigma =C{k}^2(b-ak{L}_{\mathrm{sat}})/(1+k^2{L}_{\mathrm{sat}}^2)$. Thus $b>0$ permits long-wave growth when $a>0$, while a finite saturation length stabilizes shorter waves.

For a small sinusoidal bedform, a fluid-dynamical closure can be written $\hat{\tau }={\tau }_{0}k(A+iB)\hat{\eta }$ for $k>0$. The real coefficient $A$ is in phase with the bed height and $B$ describes an upstream phase lead. Neither coefficient follows from the Exner equation or sediment relaxation alone; they require a flow calculation or measurement.

An acoustic wave is a propagating disturbance in a material's pressure, mass density and particle motion. Compressing a small fluid region raises its pressure; the resulting pressure gradient accelerates adjacent material, while inertia carries the motion onward. This restoring mechanism permits longitudinal waves in an ordinary fluid.

For small perturbations about a uniform resting state, the linear continuity equation and momentum conservation combine into a wave equation. The squared speed of sound is the relevant derivative of pressure with respect to mass density; rapid locally reversible compression usually uses the isentropic derivative. A microphone records the pressure variation, while a travel-time measurement probes propagation speed. The particle velocity can remain much smaller than the speed of sound: local material oscillation and movement of the wave pattern are different quantities. Acoustic energy flux relates the pressure and particle-velocity fields to transported power.

For a rigid cylinder of radius $a$ and height $h$, the velocity potential obeys the wave equation with Neumann boundary conditions. Separation in cylindrical coordinates gives radial factors $J_m(k_{m,n}r/a)$, angular sine and cosine factors of integer order $m$, and axial factors $\cos (\ell \pi z/h)$. The nonzero radial numbers satisfy $J_m^{\prime }(k_{m,n})=0$, yielding the displayed frequencies.

There is also a constant radial mode, with $m=0$ and radial number zero. A formal zero of $J_m^{\prime }$ at the origin for higher $m$ must not be counted by substituting $k=0$ into $J_m(kr)$, which would give the zero function. Solving the zero-radial-eigenvalue equation directly leaves only the constant mode under the side-wall condition. The fully constant potential has zero velocity and is a gauge mode; constant radial modes with nonzero axial index are physical longitudinal oscillations.

Nonlinear acoustics retains finite-amplitude effects in sound waves, including amplitude-dependent propagation speed, distortion and possible shock waves. For a one-dimensional polytropic flow with $p\propto {\rho }^{\gamma }$, a rightgoing simple wave has characteristic speed $u+c=c_0+(\gamma +1)u/2$ relative to a quiescent reference. Thus small-amplitude linear acoustics need not be uniform when a background velocity nearly cancels the speed of sound.

Ultrasound is sound above the conventional upper limit of human hearing, approximately $20\mathrm{kHz}$. In a crystal, ultrasonic travel-time measurements determine long-wavelength elastic wave speeds, which are the small-wavevector slopes of the corresponding acoustic phonon branches.

Acoustic measurement of a moving-scatterer velocity projection using frequency or phase changes. A speed estimate needs the scattering geometry; finite pulsed sampling can alias the measured shift.

An emitted acoustic pulse is followed by detection of returning echoes. Travel time estimates depth under a declared sound-speed and path model, while echo amplitudes depend on interfaces and propagation losses.

An M-mode ultrasound scan repeatedly samples one pulse-echo ultrasound beam and places successive depth profiles alongside one another in time. Depth is one display axis and acquisition time the other; echo strength controls brightness. This makes motion along the selected line visible. ^[ref]

A stationary interface traces a line at constant depth; an oscillating interface traces a varying depth. Converting echo delay to depth still assumes a sound speed and a return path. This display records one beam over time, whereas a B-mode ultrasound scan combines beam lines across space.

A B-mode ultrasound scan maps echo strength to displayed brightness and assembles multiple pulse-echo ultrasound lines into a spatial image. Each echo's location is inferred from beam direction and return time using an assumed sound speed. ^[ref]

A bright point represents a strong received echo, not a direct measurement of tissue density. Acoustic impedance contrasts, propagation losses and the instrument's display mapping all affect brightness. A moving image consists of repeated spatial frames; in an M-mode ultrasound scan, the second display axis instead tracks time along one beam.

An A-mode ultrasound scan plots the returning echo amplitude along one pulse-echo ultrasound beam. With assumed sound speed $c$, echo delay $\tau$ is converted to depth by $z=c\tau /2$; the factor two accounts for outward and return travel. Peaks mark reflecting interfaces along that line. ^[ref]

The horizontal coordinate is depth inferred from travel time, while the vertical coordinate is signal amplitude. An A-mode ultrasound scan therefore provides a one-dimensional depth profile rather than the spatial cross-section assembled by a B-mode ultrasound scan.

For a pure rightgoing acoustic wave, retarded time $\theta =t-x$ and slow distance $X=Mx$ separate fast oscillations from nonlinear steepening. Expanding the equation of state gives $p=\rho +M(\gamma -1){\rho }^2/2+O(M^2)$. The leading perturbations satisfy $u_0={\rho }_0=p_0$. Adding the first-order mass conservation and momentum conservation equations cancels the unknown first corrections and yields the displayed viscous Burgers equation. Inviscid characteristics satisfy $\theta =s-(\gamma +1)F(s)X/2$, $p=F(s)$; loss of invertibility marks shock formation and the end of the smooth inviscid approximation.

A plane-parallel atmosphere with isothermal sound speed $c_s$ in gravity $-{\Omega }^{2}z$ has mass density ${\rho }_0{e}^{-z^2/(2H^2)}$, $H=c_s/\Omega$. Above a rigid base, vertically directed adiabatic fluid perturbations satisfy $w^{\prime \prime }-z{w}^{\prime }/H^2+({\sigma }^2/{\Omega }^2-1)w/(\gamma H^2)=0$ and $w(0)=0$. Set $x=z/H$ and $\lambda =({\sigma }^2/{\Omega }^2-1)/\gamma$. The substitution $y=e^{-x^2/4}w$ converts finite Gaussian-weighted kinetic energy into square integrability for $-y^{\prime \prime }+x^{2}y/4=(\lambda +1/2)y$. Odd extension across the base and the oscillator lowering operator force $\lambda$ to be an odd positive integer. Thus $w_n$ is proportional to the Probabilists' Hermite polynomial ${\mathrm{He}}_{2n+1}(z/H)$. These are vertical acoustic waves; a general stably stratified atmosphere also admits internal gravity waves with horizontal structure.

An acoustically rigid boundary has zero perturbation normal velocity. For a reflected acoustic plane wave, pressure amplitudes add in phase and the reflection coefficient is $+1$. It is the infinite-surface acoustic impedance limit. A pressure-release boundary instead fixes the pressure perturbation to zero and reflects with coefficient $-1$.

A stratified atmosphere can reflect low-frequency acoustic waves while permitting sufficiently high-frequency waves to escape. In a plane-parallel isothermal atmosphere, the angular acoustic cutoff is approximately $c_s/(2H_{\rho })$, where $H_{\rho }$ is the density scale height. Its detailed form depends on the atmospheric profile and on the wave-variable approximation.

In an isothermal atmosphere with constant density scale height $H$, the equation $\delta {\rho }_{tt}-c^2\delta {\rho }_{zz}-(c^2/H)\delta {\rho }_z=0$ loses its first spatial derivative under $\delta \rho =e^{-z/(2H)}\Psi$. Direct differentiation gives ${\Psi }_{tt}-c^2{\Psi }_{zz}+c^2\Psi /(4H^2)=0$. Thus a plane wave has dispersion relation ${\omega }^2=c^2{k}^2+{\omega }_c^2$, with real acoustic cutoff frequency ${\omega }_c=c/(2H)$. The cutoff term is positive; a minus sign would instead predict growing sufficiently long spatial modes. Vertical propagation requires $|\omega |\ge {\omega }_c$ within this ideal constant-coefficient model.

A pressure-release boundary imposes zero acoustic pressure disturbance, $\tilde{p}=0$. The incident and reflected pressure amplitudes therefore obey $1+R=0$, so the pressure reflection coefficient is $R=-1$. It contrasts with a rigid inviscid boundary, where zero normal velocity gives $R=+1$. A much lower normal impedance on the far side approximates the pressure-release condition.

A sonic boom is the sound associated with shock waves from a body moving at a supersonic Mach number. The weak-wave Mach cone describes a propagation geometry rather than the finite pressure signature determining the sound strength.

A shear flow has a velocity field varying across adjacent fluid layers, giving a nonzero shear rate. A parallel example is $\mathbf{u}=U(z){\mathbf{e}}_x$, with vertical shear $U^{\prime }$. Linear shear flow has constant shear, while a shear layer concentrates the velocity change in a narrow region.

Variation of horizontal velocity with height. In a rotating balanced stratification, thermal-wind balance relates this vertical shear to a horizontal buoyancy gradient. The sign reverses when changing from upward height to downward-increasing pressure coordinate.

A renovating shear flow holds a planar shear flow fixed for an interval $T$, then chooses a fresh orientation independently of preceding intervals. Each interval acts on a material line by a rotated copy of the displayed deformation gradient. The orientation law must be specified: independent uniformly distributed orientations define the isotropic model, whereas a biased orientation law generally gives another Lyapunov exponent.

For independent uniform orientations in a renovating shear flow, the incident angle relative to each new shear is uniform even after earlier deformations. One interval stretches a unit material line by $\ell (\theta {)}^2=1+a\sin 2\theta +a^2{\sin }^2\theta$. Write this as $A+B\cos (2\theta -\delta )$, where $A=1+a^2/2$ and $A^2-B^2=1$. The angular logarithmic integral gives $⟨\log \ell ⟩=\frac{1}{2}\log [(A+1)/2]$. Add these independent logarithmic increments to get the displayed rate. At fixed shear rate, small $T$ gives $\mu \sim {\Lambda }^{2}T/8$ and large $T$ gives $\mu \sim \log (|\Lambda |T/2)/T$. The unique positive maximum is at $|\Lambda |T\approx 3.9605826$, obtained from $\log z=2(1-z^{-1})$ with $z=1+a^2/4$.

The logarithm of the stretch after $N$ independent uniform renovations is a sum of independent identically distributed bounded increments. For singular stretching $e^{\gamma }$, $\gamma =\mathrm{arsinh}(|a|/2)$, the increment has Fourier expansion $r+{\sum }_{n\ge 1}(-1{)}^{n+1}{q}^n\cos (2n\theta )/n$, with $q=\tanh \gamma$ and $r=\frac{1}{2}\log (1+a^2/4)$. Orthogonality proves the displayed variance. The central limit theorem gives an approximate central log-normal distribution of ${\lambda }_N$, not an exact description of extreme tails. For $|a|\ll 1$, both the mean and variance of $\log {\lambda }_N$ are $N{a}^2/8$ to leading order. Direct expansion of fixed-power moments gives $⟨{\lambda }_N^p⟩=\exp [Np(p+2)a^2/16+O(N{a}^4)]$; in particular mean stretch grows faster than typical stretch.

A linear shear flow has velocity that varies linearly in a transverse coordinate, such as $\mathbf{u}=-Sx{\mathbf{e}}_y$. Its velocity gradient and shear rate are spatially constant.

Drag is the component of fluid force opposing an object's motion relative to the surrounding fluid. Surface pressure and viscous shear stress both contribute; their relative importance depends on shape and flow regime. A body can therefore experience drag even if most resistance comes from an uneven pressure field rather than direct tangential rubbing.

At low Reynolds number, the Stokes drag law gives $F=6\pi \mu aU$ for a sphere under its creeping-flow assumptions. In many inertial regimes, a useful representation is $F=C_D\rho A{U}^2/2$, with a measured or modeled drag coefficient $C_D$. That coefficient can vary with Reynolds number, surface condition and geometry. Towing a body while measuring force at several speeds tests which regime applies. The supplied towing power is $FU$ for steady motion along the drag direction.

Lift-induced drag is the component of aerodynamic drag associated with the three-dimensional flow that supports a finite wing’s lift. The trailing vorticity produces downwash, changing the local relative-flow direction. A force approximately perpendicular to that local flow consequently has a component opposing the freestream motion. NASA’s primary account identifies the counter-rotating wingtip vortices and the downstream force component produced by their induced flow.^[ref]

For a small induced angle ${\alpha }_i$, a force $L^{\prime }$ perpendicular to the local flow has freestream components $L=L^{\prime }\cos {\alpha }_i$ and $D_i=L^{\prime }\sin {\alpha }_i$. Hence $D_i/L=\tan {\alpha }_i\approx {\alpha }_i$. This elementary projection explains why changing the flow direction can change drag even without adding direct viscous rubbing to that model.

A useful finite-wing model iswhere $b$ is wingspan, $S$ the wing reference area and $e$ the span efficiency factor. Use the same reference area in $C_L=L/(q_{\infty }S)$ and $C_{D_i}=D_i/(q_{\infty }S)$, with $q_{\infty }=\rho U^2/2$. NASA gives this coefficient relation and aspect-ratio definition.^[ref] It is a model of the induced contribution, not a statement that total drag has only this dependence in every flow regime.

An original fixed-lift calculation eliminates the coefficients:Take $L=6000\mathrm{N}$, $b=10\mathrm{m}$, $S=12.5{\mathrm{m}}^2$, $e=0.8$, $\rho =1.2kg/{\mathrm{m}}^3$ and $U=30m/s$. Then $AR=8$, $q_{\infty }=540\mathrm{Pa}$, $C_L\approx 0.8889$, and $D_i\approx 265.3\mathrm{N}$. The corresponding induced-power contribution is $D_{i}U\approx 7.96\mathrm{kW}$. Doubling the speed while holding lift, span, density and efficiency fixed reduces this modeled drag to about $66.3\mathrm{N}$.

What is held fixed matters. At fixed $C_L$ instead of fixed lift, induced drag grows with dynamic pressure. The familiar inverse-square-speed conclusion above therefore cannot be transferred to a test that holds angle of attack or coefficient fixed while changing speed. Changes in efficiency, separation or compressibility can also invalidate the assumed comparison.

Lifting-line theory relates spanwise loading, trailing vorticity and induced velocities. The ideal planar minimum for fixed lift and span has elliptic span loading; this is a statement about the load distribution, not merely the outline of the wing. Total drag measurements also contain profile and other contributions, so fitting a total-drag curve does not automatically isolate $D_i$.

A proposed wind-tunnel study measures force components and freestream conditions for several lift settings, while separately accounting for support loads, wall effects and profile drag. A downstream velocity survey tests the predicted downwash structure. Force balance alone observes total aerodynamic loads; the split into induced and other drag requires the additional flow information or a justified model. Report those assumptions before inferring an efficiency factor from the measured drag.

Downwash is a downward component of fluid velocity, commonly the induced flow associated with a lifting wing and its trailing vortices. For positive upward lift, this flow changes the direction of the local relative wind and contributes to lift-induced drag. NASA's primary explanation connects the trailing vortex system, reduced effective incidence and rearward component of the aerodynamic force.^[ref]

Choose streamwise velocity $U>0$ and let $w>0$ denote downward induced speed at a section. The induced angle and effective angle of attack areAn ideal local lifting force normal to that relative flow is tilted rearward. Its vertical and downstream projections obey $D_i/L=\tan {\alpha }_i=w/U$; hence $D_i\approx L{\alpha }_i$ for small induced angles. This projection isolates the induced contribution in the stated model; it does not include skin friction or every source of total drag.

For an original uniform-downwash example, take $U=50m/s$, $w=2m/s$ and total lift $L=10000\mathrm{N}$. Then ${\alpha }_i\approx 2.291^{\circ }$ and $D_i=400\mathrm{N}$ in the projection model. The power required to overcome this contribution is $U{D}_i=20\mathrm{kW}$. At geometric incidence $6^{\circ }$, the effective incidence is about $3.709^{\circ }$.

A real finite wing generally has span-dependent loading and induced speed. In small-angle lifting-line theory,Using an unweighted arithmetic mean of measured $w$ need not reproduce that integral. For elliptic span loading, the ideal lifting-line downwash is constant across the span and ${\alpha }_i\simeq C_L/(\pi AR)$, consistent with $C_{D_i}=C_L^2/(\pi AR)$. These statements concern the ideal planar model rather than arbitrary wakes or separated flow.

A proposed observation measures flow direction and speed at a specified plane around a loaded wing, using calibrated directional probes or timed tracer images. Repeat with zero and reversed lift, and compare the sign and spatial distribution of the induced flow. Record the freestream reference and wing incidence as well as the probe positions. Wake-plane dependence of a downwash measurement matters: a velocity measured well downstream cannot simply be inserted as the wing-section induced speed without a model connecting the two planes. Visible tracer deflection demonstrates flow direction only to the extent that the tracer follows the fluid; it is not by itself a quantitative lift or drag measurement.

Wake-plane dependence of a downwash measurement means that a measured downward velocity depends on where the observation plane lies relative to the lifting wing and its evolving wake. The induced velocity entering a wing-section force calculation is not generally equal to a downstream probe reading. Classical lifting-line theory explicitly adjusts local incidence using the induced flow at the lifting line, under assumptions such as steady flow, incompressible flow and a nearly planar high-aspect-ratio wing.^[ref]

An original straight-filament model isolates one reason for the difference. Consider a vortex filament of circulation magnitude $\Gamma$ starting at streamwise coordinate zero and extending downstream forever. At a point with streamwise coordinate $x$ and perpendicular distance $r>0$, its induced azimuthal speed isThe integral follows by differentiating $(s-x)/[r\sqrt{(s-x{)}^2+r^2}]$. At $x=0$, only one half of the corresponding infinite straight filament contributes; far downstream, the local filament extends far in both axial directions relative to $r$.

For two straight trailing filaments at transverse positions $y=\pm a$ with opposite circulation signs chosen to induce downward flow between them, their midpoint contributions add:With stipulated $\Gamma =20{\mathrm{m}}^2/s$ and $a=5\mathrm{m}$, this trailing-filament contribution is $0.6366m/s$ at $x=0$, $1.0868m/s$ at $x=5\mathrm{m}$, and approaches $1.2732m/s$. This is an idealized geometry calculation, not a measured wake or a universal factor-of-two correction for every wing.

The model omits finite vortex cores, the bound-vortex contribution away from the lifting line, the complete distributed vortex sheet, wake deformation and wake contraction in a downwash survey. Each can change the spatial transfer. In particular, a wing-plane value and a far-wake value must be connected using the same load and wake model before using one to infer the other.

In the small-angle lifting-line model, the wing-plane induced-drag calculation weights downwash by the circulation or local lift. Its source derivation integrates the local induced contribution along the wing.^[ref] Substituting an arbitrary downstream speed into that expression can misestimate drag even if the speed measurement itself is accurate.

A proposed survey measures velocity components at several downstream stations with the same probe calibration and freestream reference, covering both vortex cores and the intervening region. Record the wing loading, probe orientation and cross-plane coordinates, and track how vortex separation and circulation estimates change. A centerline reading alone does not recover the whole momentum field. Matching an axial trend tests the proposed transfer model only over the surveyed region; it does not establish that a rolled-up real wake remains a pair of straight mathematical filaments.