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Aerodynamics

Field: Fluid Mechanics

The study of the motion of air, particularly when it interacts with a solid object.

Sequence of Expressions

Definition

Viscosity

In the context of continuum mechanics, the viscous stress tensor τ\boldsymbol{\tau} for an incompressible Newtonian fluid is defined by the relationship between the shear stress and the rate of strain. Let u(x,t)\textbf{u}(\textbf{x}, t) be the velocity field, and ρ\rho be the fluid density. The rate of strain tensor S\textbf{S} is given by the symmetric part of the velocity gradient tensor ∇u\nabla \textbf{u}: \n\nS=12(∇u+(∇u)T)\textbf{S} = \frac{1}{2} \big( \nabla \textbf{u} + (\nabla \textbf{u})^T \big) \n\nFor a Newtonian fluid, the viscous stress tensor τ\boldsymbol{\tau} is linearly proportional to the rate of strain tensor S\textbf{S}, with the proportionality constant being the dynamic viscosity ν\nu (or 12ρν\frac{1}{2}\rho\nu depending on the specific definition used, but ν\nu is standard for the coefficient): \n\nτ=2νS=ν(∇u+(∇u)T)\boldsymbol{\tau} = 2\nu \textbf{S} = \nu \big( \nabla \textbf{u} + (\nabla \textbf{u})^T \big) \n\nAlternatively, considering the shear stress component τij\tau_{ij} in Cartesian coordinates, the definition is:\n\nτij=2νDev(u)ijDev(u)ij=ν(∂ui∂xj+∂uj∂xi)\tau_{ij} = 2\nu \frac{\text{Dev}(\textbf{u})_{ij}}{\text{Dev}(\textbf{u})_{ij}} = \nu \big( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \big) \n\nThis formulation quantifies the internal resistance to deformation, where ν\nu has units of Mass⋅Length−1⋅Time−1\text{Mass} \cdot \text{Length}^{-1} \cdot \text{Time}^{-1}.
Definition

Compressibility

The compressibility of a fluid is mathematically characterized by the relationship between the pressure gradient and the density variation, quantified through the bulk modulus KK. For a fluid undergoing an adiabatic process, the speed of sound cc is defined by the thermodynamic derivative: \n\nc2=dPdρ∣adiabaticc^2 = \frac{dP}{d\rho} \bigg|_{\text{adiabatic}} \n\nAlternatively, the bulk modulus KK is defined as the ratio of the change in pressure to the resulting fractional change in volume: \n\nK=−VdPdV=ρdPdρK = -V \frac{dP}{dV} = \rho \frac{dP}{d\rho} \n\nThe reciprocal of the compressibility β\beta is thus given by the bulk modulus, β=1/K\beta = 1/K. For an ideal gas, the equation of state P=ρRTP = \rho R T yields the specific heat ratio γ=cp/cv\gamma = c_p/c_v, and the speed of sound is derived as:\n\nc=γPρc = \sqrt{\frac{\gamma P}{\rho}} \n\nThus, the compressibility β\beta is formally expressed as:\n\nβ=1c21ρdρdP∣s=const\beta = \frac{1}{c^2} \frac{1}{\rho} \frac{d\rho}{dP} \bigg|_{s=\text{const}}
The Drag Coefficient, CdC_d, is defined as the dimensionless ratio of the total drag force, DD, acting on a body BB immersed in a fluid, to the dynamic pressure multiplied by a defined reference area ArefA_{ref}. Mathematically, the drag force DD is derived from the surface integral of the stress tensor τij\tau_{ij} over the boundary surface SS of the body BB:\n\nD=∬S(σ⃗⋅n⃗)⋅i⃗flow dSD = \iint_{S} (\vec{\sigma} \cdot \vec{n}) \cdot \vec{i}_{flow} \, dS\n\nwhere σ⃗\vec{\sigma} is the Cauchy stress tensor, n⃗\vec{n} is the outward unit normal vector to SS, and i⃗flow\vec{i}_{flow} is the unit vector in the direction of the relative flow velocity v⃗\vec{v}.\n\nAssuming the flow is steady and incompressible, the drag force DD simplifies to:\n\nD=12ρv2ArefCdD = \frac{1}{2} \rho v^2 A_{ref} C_d\n\nTherefore, the rigorous definition of the Drag Coefficient is given by:\n\nCd=D12ρv2Aref=∬S(pn⃗+τ⃗)⋅i⃗flow dS12ρv2ArefC_d = \frac{D}{\frac{1}{2} \rho v^2 A_{ref}} = \frac{\iint_{S} (p \vec{n} + \vec{\tau}) \cdot \vec{i}_{flow} \, dS}{\frac{1}{2} \rho v^2 A_{ref}}\n\nHere, ρ\rho is the fluid density, vv is the characteristic flow velocity, ArefA_{ref} is the reference area (e.g., projected area), and pp and τ⃗\vec{\tau} are the pressure and viscous stress components, respectively.
The Lift Coefficient, ClC_l, is defined as the dimensionless ratio of the total lift force, LL, generated by the fluid flow acting on a body surface SS, to the dynamic pressure of the freestream flow, 12ρV2\frac{1}{2} \rho V^2, multiplied by the reference area AA. \n\nLet ρ\rho be the fluid density, VV be the freestream velocity, and AA be the reference area. The total lift force LL is obtained by integrating the pressure and shear stresses over the surface SS: \n\nL = \frac{1}{A} \bigg\{ \text{Integral}_{S} \bigg\langle (p - p_{\infty}) \bigg\rangle \normal{n} + \tau_w \bigg\rangle dS \bigg\} \n\nWhere pp is the local static pressure, p∞p_{\infty} is the freestream static pressure, τw\tau_w is the wall shear stress, and \normal{n} is the unit vector normal to the surface SS. \n\nFormally, ClC_l is given by:\n\nC_l = \frac{L}{\frac{1}{2} \rho V^2 A} = \frac{1}{\frac{1}{2} \rho V^2 A} \cdot \text{Integral}_{S} \bigg\langle (p - p_{\infty}) \normal{n} + \tau_w \bigg\rangle dS
Let u(x,t)\textbf{u}(\textbf{x}, t) be the velocity field of an incompressible Newtonian fluid (∇∙u=0\nabla \bullet \textbf{u} = 0) with density ρ\rho and kinematic viscosity ν\nu. The flow dynamics are governed by the momentum equation: \n\nρ(∂u∂t+(u⋅∇)u)=−∇p+μ∇2u+f\rho \left( \frac{\partial \textbf{u}}{\partial t} + (\textbf{u} \cdot \nabla) \textbf{u} \right) = -\nabla p + \mu \nabla^2 \textbf{u} + \textbf{f} \n\nwhere μ=ρν\mu = \rho \nu is the dynamic viscosity. \n\nWe define the characteristic scales: a length scale LL, a characteristic velocity scale U=∣∣u∣∣charU = ||\textbf{u}||_{char}, and the fluid properties ρ\rho and ν\nu. The Reynolds number, ReRe, is rigorously defined as the dimensionless ratio of the characteristic inertial forces to the characteristic viscous forces:\n\nRe=ρULμ=ULνRe = \frac{\rho U L}{\mu} = \frac{U L}{\nu} \n\nThis ratio quantifies the relative magnitude of the non-linear advective terms (inertia) versus the linear diffusive terms (viscosity) in the governing partial differential equation.
Consider a steady, incompressible flow of fluid with density ρ\rho and dynamic viscosity μ\mu over a surface defined by y=0y=0. We adopt a coordinate system (x,y,z)(x, y, z) where xx is the streamwise direction and yy is the normal direction. The flow field u⃗=(u,v,w)\vec{u} = (u, v, w) must satisfy the continuity equation and the full Navier-Stokes equations:\n\n∂u∂x+∂v∂y+∂w∂z=0\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \nρ(u∂u∂x+v∂u∂y+w∂u∂z)=−∂p∂x+μ(∂2u∂x2+∂2u∂y2+∂2u∂z2)+fx\rho (u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}) = -\frac{\partial p}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + f_x \n(and similar equations for vv and ww)\n\nBoundary Layer Theory posits that if the characteristic length scale of the boundary layer, δ(x)\delta(x), is much smaller than the characteristic length scale of the external flow, Le(x)L_e(x), i.e., δ/Le≪1\delta/L_e \ll 1, then the governing equations can be reduced via an asymptotic expansion. Assuming ∂/∂x≫∂/∂y\partial / \partial x \gg \partial / \partial y and ∂/∂z≪∂/∂x\partial / \partial z \ll \partial / \partial x, the xx-momentum equation simplifies to the boundary layer equation:\n\nρ(u∂u∂x+v∂u∂y)=−∂pe∂x+μ(∂2u∂y2)+fx\rho \left( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \right) = -\frac{\partial p_e}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial y^2} \right) + f_x \n\nSubject to the boundary conditions:\n1. No-slip condition at the wall: u(x,0)=0u(x, 0) = 0 and v(x,0)=0v(x, 0) = 0.\n2. Matching condition at the edge of the boundary layer: lim⁡y→δ(x)+(u,v)=(Ue(x),0)\lim_{y \to \delta(x)^+} (u, v) = (U_e(x), 0), where (Ue(x),0)(U_e(x), 0) is the velocity profile of the external, inviscid flow.
Euler's Equations of Motion constitute a system of partial differential equations governing the flow of a compressible, inviscid fluid (u\mathbf{u} is the velocity vector, ρ\rho is the density, pp is the pressure, and f\mathbf{f} represents external body forces per unit mass). The system is derived from the conservation laws (mass, momentum, and energy) and is expressed in the material derivative form DDt=∂∂t+u⋅∇\frac{D}{Dt} = \frac{\partial}{\partial t} + \mathbf{u} \cdot \boldsymbol{\nabla}.\n\n**1. Continuity Equation (Conservation of Mass):**\n∂ρ∂t+∇⋅(ρu)=0\frac{\partial \rho}{\partial t} + \boldsymbol{\nabla} \cdot (\rho \mathbf{u}) = 0\n\n**2. Momentum Equation (Euler Equation):**\nρ(∂u∂t+(u⋅∇)u)=−∇p+f\rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u} \right) = -\boldsymbol{\nabla} p + \mathbf{f} \n\n**3. Energy Equation (Conservation of Energy):**\nAssuming the fluid is adiabatic and no external work is done by non-pressure forces, the total energy E=e+12u⋅uE = e + \frac{1}{2} \mathbf{u} \cdot \mathbf{u} (where ee is specific internal energy) satisfies:\nρ(∂E∂t+(u⋅∇)E)=−p(∂ρ∂t+u⋅∇ρ)+u⋅fnon−pressure\rho \left( \frac{\partial E}{\partial t} + (\mathbf{u} \cdot \boldsymbol{\nabla}) E \right) = -p \left( \frac{\partial \rho}{\partial t} + \mathbf{u} \cdot \boldsymbol{\nabla} \rho \right) + \mathbf{u} \cdot \mathbf{f}_{non-pressure} \n\n*Note: For a perfect gas, the equation of state p=p(ρ,T)p = p(\rho, T) and the specific enthalpy hh are often used to close the system, relating the three equations.*
Consider two distinct bodies, AA and BB, separated by an interface Σ\Sigma. The force F⃗A→B\vec{F}_{A \to B} exerted by AA on BB is derived from the Cauchy stress tensor σA\mathbf{\sigma}_A acting across the surface Σ\Sigma, where n⃗\vec{n} is the outward unit normal vector from BB into AA. The force is given by the surface integral of the traction vector t⃗\vec{t}: \n\nF⃗A→B=∬ΣσA⋅n⃗dS=∬Σt⃗AdS\vec{F}_{A \to B} = \iint_{\Sigma} \mathbf{\sigma}_A \cdot \vec{n} dS = \iint_{\Sigma} \vec{t}_A dS \n\nSimilarly, the force F⃗B→A\vec{F}_{B \to A} exerted by BB on AA is derived from σB\mathbf{\sigma}_B acting across Σ\Sigma. The rigorous statement of Newton's Third Law in this continuum framework is the equality of these force vectors:\n\nF⃗A→B=−F⃗B→A\vec{F}_{A \to B} = -\vec{F}_{B \to A} \n\nThis implies that the traction fields must satisfy the condition: \n\n∬ΣσA⋅n⃗dS=−∬ΣσB⋅n⃗dS\iint_{\Sigma} \mathbf{\sigma}_A \cdot \vec{n} dS = -\iint_{\Sigma} \mathbf{\sigma}_B \cdot \vec{n} dS
Bernoulli's Principle is a direct consequence of the conservation of energy applied to an ideal fluid flow. Consider a steady, incompressible, and inviscid flow (ρ=constant\rho = constant, μ=0\mu = 0) along a streamline L\mathcal{L} in a conservative gravitational field. The principle states that the total mechanical energy per unit volume, PtotalP_{total}, remains constant along L\mathcal{L}.\n\nFormally, if v\mathbf{v} is the fluid velocity, PP is the static pressure, ρ\rho is the fluid density, and gg is the acceleration due to gravity, the conservation of energy dictates:\n\nPρ+12∣v∣2+gh=Cstreamline\frac{P}{\rho} + \frac{1}{2} |\mathbf{v}|^2 + g h = C_{streamline} \n\nwhere hh is the elevation (potential energy per unit weight), and CstreamlineC_{streamline} is a constant value for all points on the streamline. \n\nFor two points, 1 and 2, along the streamline, the principle is expressed as:\n\n(P1+12ρ∣v1∣2+ρgh1)=(P2+12ρ∣v2∣2+ρgh2)\left(P_{1} + \frac{1}{2} \rho |\mathbf{v}_{1}|^2 + \rho g h_{1}\right) = \left(P_{2} + \frac{1}{2} \rho |\mathbf{v}_{2}|^2 + \rho g h_{2}\right) \n\nThis formulation requires the flow to satisfy the Euler equations and the continuity equation, ensuring that the work done by pressure forces and gravity is balanced by the change in kinetic energy.
The conservation of mass principle, applied to a continuous fluid flow within a control volume Ω(t)⊂R3\Omega(t) \subset \mathbb{R}^3, is mathematically expressed by the continuity equation. Assuming the fluid is described by a density function ρ(x⃗,t)\rho(\vec{x}, t) and a velocity field u⃗(x⃗,t)\vec{u}(\vec{x}, t), the statement is:\n\n∂ρ∂t+∇⋅(ρu⃗)=0in Ω(t)\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0 \quad \text{in } \Omega(t)\n\nThis equation asserts that the rate of change of density within the control volume (the first term) must be exactly balanced by the net flux of mass out of the volume (the divergence of the mass flux, ∇⋅(ρu⃗)\nabla \cdot (\rho \vec{u})). Alternatively, using the Reynolds Transport Theorem (RTT) for a fixed control volume VV, the conservation law states that the rate of change of mass within VV equals the net mass flux across the boundary surface S=∂VS = \partial V:\n\nddt∫Vρ dV=−∮Sρu⃗⋅n⃗ dS\frac{d}{dt} \int_{V} \rho \, dV = - \oint_{S} \rho \vec{u} \cdot \vec{n} \, dS\n\nHere, n⃗\vec{n} is the outward-pointing unit normal vector to the surface SS. The differential form, ∂ρ∂t+∇⋅(ρu⃗)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0, is the most fundamental statement for local conservation.