空力弾性解析
Theory and Physics
What is Aeroelasticity?
Professor, aeroelasticity is a coupled problem of aerodynamics and structures, right? Why is it considered particularly important?
Aeroelasticity is explained by Collar's triangle. It's a problem where three forces combine: Aerodynamic Forces, Elastic Forces, and Inertial Forces. The interaction of these three forces causes dangerous phenomena like flutter, divergence, and buffeting.
Especially flutter is a self-excited vibration caused by the coupling of aerodynamic and structural elastic forces, where the amplitude grows exponentially leading to structural failure. In aircraft design, proving that flutter does not occur within the flight envelope is a mandatory requirement for type certification.
2-Degree-of-Freedom Flutter Model
Please explain the basic mechanism of flutter.
A typical airfoil flutter is explained by a 2-degree-of-freedom model of bending (plunge, $h$) and torsion (pitch, $\alpha$). The equations of motion are:
Here $m$ is mass, $S_\alpha$ is the static unbalance moment, $I_\alpha$ is the moment of inertia, $K_h$, $K_\alpha$ are spring constants, $L$ is lift, and $M_{ea}$ is the moment about the elastic axis.
How is the condition for flutter onset determined?
Since unsteady aerodynamic forces are a function of velocity, as the velocity increases, the frequencies of the bending and torsion modes approach each other (frequency coalescence), and at a certain velocity, the net energy transfer becomes positive, causing the vibration to diverge. This critical velocity is the flutter speed $V_F$.
Here $\sigma$ is the eigenvalue of the system.
Theodorsen's Unsteady Aerodynamic Theory
How are unsteady aerodynamic forces calculated?
Classically, the Theodorsen function $C(k)$ is used. The unsteady lift for a harmonically oscillating airfoil is:
Here $k = \omega b / U$ is the reduced frequency, $b$ is the semi-chord length, and $a$ is the elastic axis position. $C(k)$ is expressed using Bessel functions, converging to $C \to 1$ for $k \to 0$ (quasi-steady) and $C \to 0.5$ for $k \to \infty$.
So if we use CFD, we don't need to rely on the Theodorsen function, right?
Exactly. In CFD-based aeroelastic analysis, unsteady aerodynamic forces are calculated directly from CFD, enabling predictions beyond the applicability limits of theoretical models (linear, potential flow), such as transonic flutter and large-amplitude oscillations.
The Lockheed L-188 Disaster Changed Aeroelasticity
In 1960, a series of Lockheed L-188 Electra aircraft disintegrated in mid-air due to propeller resonance flutter. The investigation revealed that a design flaw in the propeller's vibration damper changed the natural frequency, inducing flutter under certain flight conditions. Following this accident, the submission of aeroelastic flutter analysis became mandatory for aircraft type certification in the United States. This is a classic example where an accident forced the implementation of theory, and flutter theory is now ingrained as a fundamental rule in aircraft design.
Physical Meaning of Each Term
- Time term $\partial(\rho\phi)/\partial t$: Imagine turning on a faucet. At first, water comes out spluttering and unstable, but after a while, the flow becomes steady, right? This "during the change" is described by the time term. The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens and closes, are all unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning this term is set to zero. Since computational cost drops significantly, solving first in steady-state is a basic CFD strategy.
- Convection term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow becomes faster, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is transport by flow, conduction is transfer by molecules. There's an order of magnitude difference in efficiency.
- Diffusion term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while it naturally mixes, right? That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is large, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flow (slow, viscous), diffusion is dominant. Conversely, in high Re number flow, convection overwhelmingly dominates, and diffusion plays a supporting role.
- Pressure term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle tip, right? Why? Because the piston side is high pressure, the needle tip is low pressure—this pressure difference becomes the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Where there is a pressure difference, flow is generated"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, mixing up absolute/gauge pressure might be the cause.
- Source term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed up by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat generated by a gas stove flame, Lorentz force applied to molten metal by a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, if you forget to include buoyancy, the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a room with the heater on in winter.
Assumptions and Applicability Limits
- Continuum assumption: Valid for Knudsen number Kn < 0.01 (molecular mean free path ≪ characteristic length)
- Newtonian fluid assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility assumption (for Ma < 0.3): Density is treated as constant. For Mach numbers above 0.3, compressibility effects must be considered.
- Boussinesq approximation (natural convection): Density variation is considered only in the buoyancy term, using constant density in other terms.
- Non-applicable cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (VOF/Level Set, etc. required)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units. |
| Pressure $p$ | Pa | Distinguish between gauge pressure and absolute pressure. Use absolute pressure for compressible analysis. |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C |
| Viscosity coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds number $Re$ | dimensionless | $Re = \rho u L / \mu$. Indicator for laminar/turbulent transition. |
| CFL number | dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability. |
Numerical Methods and Implementation
Classification of CFD-Based Aeroelastic Analysis
What approaches are there for aeroelastic analysis using CFD?
The most commonly used in practice is CFD + Modal Analysis. The structure is represented by an expansion in its natural modes, and the time evolution of the generalized coordinates $q_i(t)$ for each mode is solved coupled with unsteady aerodynamic forces from CFD.
Here $Q_i$ is the generalized aerodynamic force calculated from CFD. A standard workflow is to obtain structural modes from MSC Nastran SOL 146 (Flutter Analysis) and pass those mode shapes to the CFD solver.
How is data transferred between Nastran and the CFD solver?
Using Ansys System Coupling, mesh displacement and surface pressure can be automatically mapped between Fluent (fluid) and Ansys Mechanical (structure). For coupling with Nastran, you can build a custom workflow via Fluent UDF to pass modal coordinates, or use dedicated tools (e.g., MSC FlightLoads, Zona ZAERO).
V-g Method and p-k Method
What is the standard method for determining flutter speed?
The v-g method (velocity-damping method) and p-k method for linear flutter analysis are the basics.
V-g Method: Assumes harmonic oscillation at each velocity and determines the required structural damping $g$. The velocity where $g = 0$ is crossed is the flutter speed. Nastran's SOL 145 uses this method.
p-k Method: Determines eigenvalues in the time domain and plots damping ratio and mode frequency as functions of velocity. The p-k method gives physically more correct damping information and can also be used for damping estimation in the subcritical region. Nastran's SOL 145 PK option.
In CFD-based flutter analysis, can't these classical methods be used?
For CFD, the most direct method is the "time-marching method," which calculates the unsteady time history directly and observes whether the response damps or diverges. The flutter boundary is identified by incrementally increasing the velocity parameter. However, since computational cost is high, it's efficient to first narrow down the approximate range using linear theory (DLM: Doublet Lattice Method + Nastran SOL 145/146) and then refine with CFD.
DLM and CFD Correction
Is DLM (Doublet Lattice Method) still used?
DLM is still the mainstay for aircraft flutter certification. It can efficiently calculate unsteady aerodynamic forces in the frequency domain under potential flow assumptions. However, it cannot handle shock wave effects in the transonic regime, so "CFD-corrected DLM," which corrects DLM aerodynamic forces using steady pressure distributions calculated by CFD, is widely used.
"Method Selection" for Aeroelastic Analysis Has Always Been a Headache
Before CFD-based aeroelastic analysis became widespread, designers used flutter calculations combining linear panel methods and structural modes. Accuracy was rough but computation was fast. Using CFD improves accuracy but skyrockets cost. Records remain of an Airbus development team seriously debating in the late 1990s "how much CFD should be used," concluding that "nonlinear CFD is unnecessary if the required accuracy is within ±5% of the flutter speed." The importance of having criteria for method selection remains unchanged today.
Upwind Differencing (Upwind)
1st-order upwind: Large numerical diffusion but stable. 2nd-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
2nd-order accuracy, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Maintain high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shock waves and steep gradients.
Finite Volume Method vs Finite Element Method
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