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In spite of the inherent difficulty, reproducing the exact structure of real flows is a critically important issue in many fields, such as weather forecasting or feedback flow control. In order to obtain information on real flows, extensive studies have been carried out on methodology to integrate measurement and simulation, for example, the four-dimensional variational data assimilation method (4D-Var) or the state estimator such as the Kalman filter or the state observer. Measurement-integrated (MI) simulation is a state observer in which a computational fluid dynamics (CFD) scheme is used as a mathematical model of the physical system instead of a small dimensional linear dynamical system usually used in state observers. A large dimensional nonlinear CFD model makes it possible to accurately reproduce real flows for properly designed feedback signals. This review article surveys the theoretical formulations and applications of MI simulation. Formulations of MI simulation are presented, including governing equations of a flow field observer, those of a linearized error dynamics describing the convergence of the observer, and stabilization of the numerical scheme, which is important in implementation of MI simulation. Applications of MI simulation are presented ranging from fundamental turbulent flows in pipes and Karman vortices in a wind tunnel to clinical application in diagnosis of blood flows in a human body.

Recent advances in computational fluid dynamics (CFD) enable calculation of complex flows appearing in many practical applications with reasonable accuracy. However, an accurate solution usually does not mean a solution that reproduces the exact instantaneous structure of the real flow such as a turbulent flow, but rather one having the same statistical characteristics as those of the relevant flow [

In spite of the inherent difficulty, reproducing the exact structure of real flows is a critically important issue in many fields, such as weather forecasting or feedback flow control. In numerical simulation used for weather forecasting, the initial condition is updated at time intervals based on past computational results and measurement data around the computational grid points. In meteorology, extensive studies have been carried out on such methods, which are termed data assimilation [

Measurement-integrated simulation is a state observer for a flow field. An observer is a common tool in control theory to estimate the real state from a mathematical model and partial measurement [

other existing observers, is the usage of a CFD scheme as a mathematical model of the physical flow. A large dimensional nonlinear CFD model makes it difficult to design the feedback law in a theoretical manner; therefore, it has been determined by a trial-and-error method based on physical considerations. However, it makes it possible to accurately reproduce real flows once the feedback law is properly designed.

This review article deals with the theoretical background and applications of MI simulation. In Section 2, formulations of MI simulation are presented, including governing equations of an observer for a flow field, those of a linearized error dynamics describing the convergence of the observer [

Measurement-integrated simulation deals with incompressible and viscous fluid flow. The dynamic behavior of the flow field is governed by the Navier-Stokes equation:

and the equation of mass continuity:

as well as by the initial and boundary conditions. In the Navier-Stokes Equation (1), p denotes the pressure divided by density, and f denotes the body force divided by density which is defined as the feedback signal in the MI simulation. The pressure equation is derived from Equation (1) and (2) as

We use Equations (1) and (3) as the fundamental equations.

The basic equation of the MI simulation is represented as a spatially discretized form of governing equations (1) and (3):

where u_{N} and p_{N} are computational results for the 3N-dimensional velocity vector and the N-dimensional pressure vector, respectively, N denotes the number of grid points, g_{N}, q_{N}, ∇_{N} and Δ_{N} are matrices which express the discrete form of operators g, q, ∇, and Δ. The operators, g and q are defined as follows

It is noted that effects of the boundary conditions are included in the functions g_{N} and q_{N}.

We define the operator _{N} is naturally extended to the case when the variable is a velocity vector field as

On the other hand, we apply external force denoted by a function of real flow and numerical simulation in MI simulation. In this study, we consider the case in which external force f_{N} is denoted by a linear function of the difference of velocity and pressure between real flow and numerical simulation:

where K_{u} denotes the 3N-by-3N feedback gain matrix of velocity, K_{p} denotes the 3N-by-N feedback gain matrix of pressure, C_{u} and C_{p} denote the 3N-by-3N and N-by-N diagonal matrices consisting of diagonal elements of 1 for measurable points or 0 for immeasurable points, and 3N-dimensional vector ε_{u} and N-dimensional vector ε_{p} mean measurement error. By introducing Equation (6) into Equation (4), we derive the general formulation of MI simulation:

We now derive the linearized error dynamics of MI simulation. Disregarding the second order and higher order terms in Taylor expansion for the difference between real flow and the basic equation of MI simulation with respect to u_{N} ‒ D_{N}(u) and p_{N} ‒ D_{N}(p), we can derive the linearized error dynamics:

and complementary static equation for pressure error:

where the underlined terms are caused by the model error, including that in the boundary conditions, and the double-underlined terms are caused by measurement error.

Here, we derive the basic equation of eigenvalue analysis for the linearized error dynamics in Equations (8) and (9), considering the case of no model error, including that in the boundary conditions, no measurement error, and feedback with only velocity components (K_{p} = 0). In this case, Equation (8) is written as

Next we reduce the dimension of the velocity error vector e_{u} based on the Weyl decomposition. In Weyl decomposition, any vector field w can be uniquely decomposed into the orthogonal vector fields v and grad ϕ as

where n denotes the unit vector normal to the boundary. In the present analysis, the velocity error e_{u} consists only of v component in Weyl decomposition since it satisfies the divergence-free condition and it vanishes on the boundary due to the above mentioned assumption of no model error. This enables us to reduce the dimension of e_{u} corresponding to that of the component of grad

We define

The projection of Equation (10) onto

We can analyze the linearized error dynamics from the eigenvalues of the 2N-by-2N system matrix

This section deals with the destabilization phenomenon of MI simulation [

We consider the case in which the time derivative term of Equation (4) is discretized with the first order implicit scheme. In this case, considering Equation (6) with

where (‒1) of the second component of the subscript of the left-hand side represents the value of the former time step. In the present study, we employ the SIMPLER method [

Since the first term of the right-hand side of Equation (13) is nonlinear with respect to

The above linear equation for

Equation (13). It is noted that the feedback signal is included in Γ as the source terms in the formulation of MI simulation of former studies.

Taking the difference between Equation (14) evaluated at iteration number n and that evaluated at n ‒ 1, and applying the mean value theorem, the following relation is obtained:

Norms of matrices in the above expression are defined as the induced norms which represent the maximum magnification of linear transformation [

From Equation (15), a sufficient condition of convergence of

cient in the iteration is less than 1. Substituting the second and third expressions of Equation (14) into this relation, we obtain the following relation:

where

Assuming that the computational time step Δt is relatively small and that the feedback gain is relatively large, we obtain the sufficient condition of convergence of MI simulation as

This relation agrees with the empirical relation obtained in a former study [

In summary, the destabilization phenomenon of existing MI simulation is ascribed to the fact that a large feedback signal in the source term determines the leading term of the numerator of the coefficient for the successive change in the iterative calculation, and, therefore, divergence occurs with a coefficient larger than 1 or with a feedback gain larger than the critical value.

Patankar [

According to the above discussion, it is naturally expected that the cause of the destabilization phenomenon can be eliminated by evaluating the feedback signal in the linear term in the iterative calculation. If we move the feedback term relating to

where modification from the original scheme is underlined. Using the above expressions, we obtain the following relation instead of Equation (16).

This relation is satisfied for the feedback gain matrix having only positive or zero eigen values, which is always satisfied for appropriate MI simulation.

The validity of the stabilized MI simulation scheme is investigated for the case of the turbulent flow in a square duct.

In this section, applications of MI simulation are presented ranging from fundamental turbulent flow in pipes [

Hayase et al. [

square duct showing that the observer, being a fundamental element in the control system theory, is also of potential use in the analysis of flow related problems as an integrated computational method with the aid of experimental measurement. A standard finite volume flow simulation algorithm was modified to the observer by adding a feedback controller, which compensated the boundary condition of the simulation based on the estimation error between output signals of the experimental measurement and the computational result. In that study, the validity of the proposed observer was confirmed by numerical experiment for the turbulent flow through a duct with a square cross section. The physical flow was modeled by a pre-calculated numerical solution of the developed turbulent flow. The estimation error in the stream-wise velocity component at the grid points on the output measurement plane was fed back to the pressure boundary condition based on the simple proportional control law.

where K_{p} is the proportional feedback gain,

Reproduction of a turbulent flow in a square duct by MI simulation was further investigated by Imagawa et al. [

Nakao et al. [

Nisugi et al. [

back signal to compensate for the error in the pressure on the side walls of the cylinder and the feed-forward signal to adjust the upstream velocity boundary condition. As compared with the ordinary simulation, the hybrid wind tunnel substantially improves the accuracy and efficiency of the analysis of the flow. Especially, the oscillation of the flow due to the Karman vortex street reproduced with the hybrid wind tunnel exactly synchronizes with that of the experiment. In comparison with the experimental measurement, the hybrid wind tunnel provides more detailed information on the flow than the experiment.

Yamagata et al. [

Acquisition of detailed information on the velocity and pressure fields of blood flow is essential for development of an accurate diagnosis or treatment for circulatory diseases. A possible way to obtain such information is integration of numerical simulation and color Doppler ultrasonography in the framework of the flow field observer. Funamoto et al. [

Ultrasonic-measurement-integrated (UMI) simulation to reproduce three-dimensional steady and unsteady blood flows in an aneurysmal aorta with realistic boundary conditions was investigated [

Recently, Kato et al. [

Magnetic resonance (MR)-measurement-integrated (MR-MI) simulation, in which the difference between the computed velocity field and the phase-contrast MRI measurement data is fed back to the numerical simulation,

has been also investigated [

In spite of the inherent difficulty, reproducing the exact structure of complex real flows is a critically important issue in many fields. In order to solve the problem, extensive studies have been carried out on methodology to integrate measurement and simulation. Measurement-integrated (MI) simulation, one of these methods, is a state observer in which a computational fluid dynamics (CFD) scheme is used as a mathematical model of the physical system. A large dimensional nonlinear CFD model makes it possible to accurately reproduce real flows for properly designed feedback signal. This review article surveyed theoretical formulations and applications of MI simulation. Formulations of MI simulation were presented for governing equations of a flow field observer, linearized error dynamics, and stabilization of the numerical scheme. Applications of MI simulation were presented ranging from fundamental turbulent flows in pipes and Karman vortices in a wind tunnel to clinical application to diagnosis of blood flows in a human body.

The author acknowledges former Transdisciplinary Fluid Integration Research Center, Advanced Fluid Information Research Center, and Institute of Fluid Science, Tohoku University for their support in establishing integrated research methodologies, including MI simulation.