13.5 Reconstruction of Data
281
13.5 Reconstruction of Data
During experiments in many fields of applied science one is confronted by the fact
that the measured quantities give only a sparse representation of the event and it is
thus difficult to describe the physical mechanism in its entirety. This lack of resolution of the data can appear in various ways. For example, in fluid mechanics
measurements usually have an insufficient resolution either in time or space so contains partial information of the flow field. Moreover, the field of exploration might be
contaminated by measurement noise or data acquisition issues. The approach called
data reconstruction originates from the field of meteorology where the weather is
forecasted by analysing the trend of the atmospheric streams and is extrapolated in
time and space, based on sparse data resulting from various types of measurement in
stations distributed around the world. Today, data reconstruction has become quite
attractive in the field of fluid mechanics as it allows the analysis of a global flow field
through limited strategic local measurements.
Data assimilation consists of a calculation-experiment coupling having for objective to fulfil the gaps of the experiment, in particular the dispersed character of
the points of measurement and their insufficient density. Digital simulations can
take form of a simple constraint (for example, to interpolate between scattered
three-dimensional vectors, under the constraint that the interpolated field respects
the incompressible Navier–Stokes equations), or of a more powerful tool of superresolution (to find the initial and boundary conditions of a numerical simulation best
approaching all the available fields of vectors during a given temporal horizon). In
addition, an important stream of research aims at the estimate of the pressure field
starting from the velocity measurements, the knowledge of the velocity components
making it possible to calculate their gradients and thus evaluating the pressure. The
method applies either in a direct way in the case of flows with low Mach number, or
by introducing additional assumptions, such as for example the isentropic character,
in the case of compressible flows.
This technique can be applied in various ways, for instance the whole flow state
(pressure, velocity, temperature, etc.) can be estimated at one given instant over the
whole spatial domain and the evolution in time can be predicted. Mathematically
this falls into the same category of inverse problem where further information is
reconstructed from a limited number of measurements. Several methods have been
developed for data reconstruction ranging from a simple interpolation to more sophisticated approaches which exploit the equations controlling the system and in fluid
mechanics the Navier–Stokes equations are employed either averaged in time or fully
unsteady. The formulation of this problem is based on the variation of parameters
approach, highly studied in meteorology.
For example, let us consider a flow which can be described by the Reynolds
Averaged Navier–Stokes equations (RANS model) formulated like classical Navier–
Stokes equations but with a model for the Reynolds stress term. This term is an
unknown and is selected as a control parameter. An algorithm of data reconstruction
can be derived by minimising the error between the mean values of that quantity
281
13.5 Reconstruction of Data
During experiments in many fields of applied science one is confronted by the fact
that the measured quantities give only a sparse representation of the event and it is
thus difficult to describe the physical mechanism in its entirety. This lack of resolution of the data can appear in various ways. For example, in fluid mechanics
measurements usually have an insufficient resolution either in time or space so contains partial information of the flow field. Moreover, the field of exploration might be
contaminated by measurement noise or data acquisition issues. The approach called
data reconstruction originates from the field of meteorology where the weather is
forecasted by analysing the trend of the atmospheric streams and is extrapolated in
time and space, based on sparse data resulting from various types of measurement in
stations distributed around the world. Today, data reconstruction has become quite
attractive in the field of fluid mechanics as it allows the analysis of a global flow field
through limited strategic local measurements.
Data assimilation consists of a calculation-experiment coupling having for objective to fulfil the gaps of the experiment, in particular the dispersed character of
the points of measurement and their insufficient density. Digital simulations can
take form of a simple constraint (for example, to interpolate between scattered
three-dimensional vectors, under the constraint that the interpolated field respects
the incompressible Navier–Stokes equations), or of a more powerful tool of superresolution (to find the initial and boundary conditions of a numerical simulation best
approaching all the available fields of vectors during a given temporal horizon). In
addition, an important stream of research aims at the estimate of the pressure field
starting from the velocity measurements, the knowledge of the velocity components
making it possible to calculate their gradients and thus evaluating the pressure. The
method applies either in a direct way in the case of flows with low Mach number, or
by introducing additional assumptions, such as for example the isentropic character,
in the case of compressible flows.
This technique can be applied in various ways, for instance the whole flow state
(pressure, velocity, temperature, etc.) can be estimated at one given instant over the
whole spatial domain and the evolution in time can be predicted. Mathematically
this falls into the same category of inverse problem where further information is
reconstructed from a limited number of measurements. Several methods have been
developed for data reconstruction ranging from a simple interpolation to more sophisticated approaches which exploit the equations controlling the system and in fluid
mechanics the Navier–Stokes equations are employed either averaged in time or fully
unsteady. The formulation of this problem is based on the variation of parameters
approach, highly studied in meteorology.
For example, let us consider a flow which can be described by the Reynolds
Averaged Navier–Stokes equations (RANS model) formulated like classical Navier–
Stokes equations but with a model for the Reynolds stress term. This term is an
unknown and is selected as a control parameter. An algorithm of data reconstruction
can be derived by minimising the error between the mean values of that quantity