Duhamel Superposition Research Articles

Unsteady Aerodynamics of Airfoils Encountering Traveling Gusts and Vortices

By means of the reverse e ow theorems, results are obtained for the unsteady lift and pitching moment on two-dimensional airfoils penetrating sharp-edged traveling gusts. Both downstream and upstream traveling gusts are considered. For the incompressible case, exact results are given and are generalized numerically for any gust e eld by means of Duhamel superposition. Results are then given for the airloads and acoustics generated by a two-dimensional airfoil encountering a vortex convecting at different gust speed ratios. Numerical results for the traveling sharp-edged gust problem are also derived for subsonic e ows by means of exact linear theory. Further results for the subsonic case are computed by means of an Euler e nite difference method. It is found that the gust speed ratio has substantial effects on the unsteady airloads and will be an important parameter to represent in helicopter rotor aeroacoustic problems.

Inverse Superposition for Calculating Food Product Temperatures during In‐container Thermal Processing

ABSTRACTA procedure was developed for theoretical calculation of internal product temperatures during in‐container sterilization of foods. Experimental product temperatures were determined through traditional heat penetration tests (processing in a retort with come‐up time of increasing temperature and holding time at constant temperature). These were used to produce, through a proposed inverse superposition solution, a “standardized,” normalized, dimensionless product temperature curve corresponding to the response of product temperature to a constant retort temperature profile. This “standardized” curve was then used, through Duhamel's superposition theorem, to calculate product temperatures for time‐varying retort temperature profiles. Limitations associated with the application of Duhamel's theorem were also transferred to the proposed methodology. Theoretical, for perfect mixing, and experimental, for conduction heating, data were used in example calculations.

Unsteady lift of a flapped airfoil by indicial concepts

A practical method is described for computing the unsteady lift on an airfoil due to arbitrary motion of a trailing-edge flap. The result for the incompressible case is obtained in state-space form by means of Duhamel superposition and employing an improved exponential approximation to Wagner's indicia! lift function. For subsonic compressible flow, the indicial lift at small values of time due to impulsive trailing-edge flap deflection is obtained from linear theory in conjunction with the aerodynamic reciprocal theorems. These results are used with experimental results for the oscillating case to obtain complete exponential approximations for the indicial response due to impulsive flap deflection. The final result for the unsteady lift due to an arbitrary flap motion in subsonic flow is obtained in state-space form. Numerical results and comparisons with experimental data are shown.

Transient response of the Trombe wall temperature distribution applicable to passive solar heating systems

The unsteady temperature distribution and heat loss history of a Trombe wall, applicable to passive solar heating, were obtained from both analytical and numerical approaches to investigate the transient performance of the Trombe wall. A one-dimensional exact solution was obtained by the Duhamel superposition technique for the time dependent boundary, and for a comparison, the numerical result was calculated through an implicit finite difference scheme. Data was obtained and discussed for the influence of solar radiation and Trombe wall thickness on the transient performance of the Trombe wall. The data may provide the thermal design basis for Trombe wall solar heating systems.

Electrochemistry of Iodide in Propylene Carbonate: II . Theoretical Model

An algorithm based on Duhamel's superposition integral is used to simulate cyclic voltammograms for iodide electrochemistry on platinum in propylene carbonate. The electrochemistry is represented by the reaction sequencewhere reaction [ I ] is rate limited (Butler‐Volmer kinetics) and reaction [ II ] is reversible (Nernst equilibrium behavior). Since the nonunity stoichiometries and Butler‐Volmer kinetics in the reaction sequence prohibit the use of analytical solution techniques, the superposition‐integral algorithm is used as a fast and accurate method for the numerical simulation. The simulated voltammograms for the iodide system agree very well with experimental observations and support the proposed reaction sequence.

The meaning of oscillations in unit hydrograph S-curves

ABSTRACT The Duhamel superposition integral is used to obtain some exact solutions for unit hydrograph applications. These equations and numerical examples are used to show that oscillations will occur in an S-curve when the time step is less than the excess rainfall duration if the measured hydrograph differs from a hydrograph that would be obtained by solving a linear differential equation with time-independent coefficients. The implications of this result with regard to the calculation of the instantaneous unit hydrograph (IUH) are discussed.

Transient response of diffusion dosimeters.

A gas sampler that operates on the principle of molecular diffusion was analyzed theoretically for its time dependent response. Applying Fick's second law of diffusion and the mathematical procedure of separation of variables and Duhamel's superposition integral, a simple technique is developed to correct concentration measurements for short term sampling and to estimate error when sampling real-time transient atmospheres.

Transient Shear Waves in Two Joined Elastic Quarter Spaces

A half space composed of two joined elastic quarter spaces of different material properties is subjected to time-varying shear tractions, which are applied parallel to the plane of juncture. The method of homogeneous solutions is used to solve the transient wave-propagation problem for the case that the surface tractions vary in time as Dirac delta functions. For surface tractions varying in time as Heaviside step functions, Duhamel superposition is employed to derive closed-form expressions for the shear stress in the plane of juncture. The interface shear stress shows a logarithmic singularity at the free surface.

Determination of undisturbed temperatures from thermocouple measurements using correction kernels

A simplified method for determining the undisturbed temperatures from thermocouple measurements is presented. The determination of the temperature which would exist if the thermocouple were not present is facilitated through the use of some correction kernels which are used in a form of Duhamel's superposition integral. Kernels are given for the case of a thermocouple embedded normal to the surface of a low thermal conductivity material.

Transient Loading of a Baffled Strip

The force-time history required to produce a unit step in the velocity of a two-dimensional plate mounted in an infinite baffle is determined and applied to more general velocity variations by Duhamel superposition.

Transient Loading of a Baffled Piston

The force-time history required to produce a step in the velocity of a circular piston mounted in an infinite baffle is determined with the aid of the Laplace transform. Working from this result, the inverse problem of the velocity response to a step force is calculated approximately. These two results suffice to treat arbitrary distributions of applied force or velocity by Duhamel superposition. The application of the results to a loudspeaker model shows that a system designed for critical damping on the “steady-state” approximation for the air loading will actually be slightly overdamped in its initial motion, but that the characteristic time during which the design assumption is inadequate is of the order of 10−3 second.