A numerical study of temperature effects on hot wire measurements inside turbulent channel flows

A way to investigate turbulence is through experiments where hot wire measurements are performed. Analysis of the in turbulence of a temperature gradient on hot wire measurements is the aim of this thesis work. Actually - to author's knowledge - this investigation is the first attempt to document, understand and ultimately correct the effect of temperature gradients on turbulence statistics. However a numerical approach is used since instantaneous temperature and streamwise velocity fields are required to evaluate this effect. A channel flow simulation at Re_tau = 180 is analyzed to make a first evaluation of the amount of error introduced by temperature gradient inside the domain. Hot wire data field is obtained processing the numerical flow field through the application of a proper version of the King's law, which connect voltage, velocity and temperature. A drift in mean streamwise velocity profile and rms is observed when temperature correction is performed by means of centerline temperature. A correct mean velocity pro�le is achieved correcting temperature through its mean value at each wall normal position, but a not negligible error is still present into rms. The key point to correct properly the sensed velocity from the hot wire is the knowledge of the instantaneous temperature field. For this purpose three correction methods are proposed. At the end a numerical simulation at Re_tau =590 is also evaluated in order to confirm the results discussed earlier.

Weighted geometric mean of large-scale matrices: numerical analysis and algorithms

Computing the weighted geometric mean of large sparse matrices is an operation that tends to become rapidly intractable, when the size of the matrices involved grows. However, if we are not interested in the computation of the matrix function itself, but just in that of its product times a vector, the problem turns simpler and there is a chance to solve it even when the matrix mean would actually be impossible to compute. Our interest is motivated by the fact that this calculation has some practical applications, related to the preconditioning of some operators arising in domain decomposition of elliptic problems. In this thesis, we explore how such a computation can be efficiently performed. First, we exploit the properties of the weighted geometric mean and find several equivalent ways to express it through real powers of a matrix. Hence, we focus our attention on matrix powers and examine how well-known techniques can be adapted to the solution of the problem at hand. In particular, we consider two broad families of approaches for the computation of f(A) v, namely quadrature formulae and Krylov subspace methods, and generalize them to the pencil case f(A\B) v. Finally, we provide an extensive experimental evaluation of the proposed algorithms and also try to assess how convergence speed and execution time are influenced by some characteristics of the input matrices. Our results suggest that a few elements have some bearing on the performance and that, although there is no best choice in general, knowing the conditioning and the sparsity of the arguments beforehand can considerably help in choosing the best strategy to tackle the problem.