Im Gegensatz zu den vorangegangenen Kapiteln befasst sich die Bioströmungsmechanik mit Strömungen, die von flexiblen biologischen Oberflächen aufgeprägt werden. Man unterscheidet die Umströmung von Lebewesen in Luft bzw. in Wasser, wie den Vogelflug oder das Schwimmen der Fische und Innenströmungen, wie z.B. der geschlossene Blutkreislauf von Lebewesen. Die Evolution hat in den vergangenen Jahrmillionen für die Fortbewegung der Lebewesen je nach Größe und Gewicht das Kriechen, Laufen, Schwimmen, Gleiten bzw. Fliegen entwickelt.

This article describes the numerical fluid-structure interaction (FSI) validation of a new pumping concept and the possibility for application of a further developed type, as an implantable ventricular assist device (VAD). The novel principle of the so-called progressive wave pump is based on the interaction of an elastic membrane actuated by forced excitation with a surrounding fluid and the pump housing. By applying forced vibrations to one end of the membrane, a transversal wave builds up and progresses to the far end generating both a positive pressure gradient and flow rate. Among others, two axisymmetric geometrical configurations are possible, namely the discoidal and the tubular design. The first one has been built as a physical prototype and is experimentally investigated. In addition, a corresponding numerical FSI model is set up and validated against the experimental findings. Based on this validated numerical method, further numerical investigations are conducted focusing on the development of a tubular progressive wave pump concept with regard to its potential for application as a VAD in the future. To address VAD-relevant issues such as size, hydraulic performance, and blood trauma, corresponding numerical simulations involving macroscopic blood trauma models have been performed. Although being still in an early phase of development, the results are promising and indicate that the wave pump concept in its present state is feasible and can be further developed and investigated as a new type of blood pump.

The effects of transport properties (viscosity and thermal conductivity) and surface roughness were compared experimentally to investigate non-self-similar Mach reflection phenomena in weak shock waves. The effect of the transport properties was changed by varying the initial pressure ahead of the incident shock in experiments conducted at Karlsruhe (KIT) and Saitama (SIT). By introducing a non-dimensional distance based on physical properties behind the Mach stem, the difference between the results under various experimental conditions was diminished, and in some cases the results almost coincide with each other within the error in angle measurement. In order to investigate the effect of surface roughness, further experiments were conducted at SIT using models with rough surfaces. The results were compared with those for smooth surfaces. The effect of surface roughness turned out to be small compared with the viscosity effect, proving the effect of transport properties to be dominant. The fact that the discrepancies between the SIT and KIT experiments vanished after introducing the non-dimensional coordinate may possibly be ascribed to inaccuracy in angle measurement at KIT.

The mass conservation at a volume element $${\rm d}V = {\rm d}x \cdot {\rm d}y \cdot {\rm d}z$$ for steady, incompressible flow $$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0,$$ with velocity components u, v,w of the velocity vector v, was introduced in Section 4.2.1. In this chapter we again consider the derivation of the continuity equation at a volume element dV , but now extended to unsteady and compressible flows. In general, the conservation of mass at a volume element may be formulated as follows: $$\begin{array}{l} The\,rate\,of\,change\,of\,mass\,in\,a\,volume\,element \\ = \sum {the\,mass\,fluxes\,into\,the\,volume\,element} \\ {\rm } - \sum {the\,mass\,fluxes\,out\,of\,the\,volume\,element.} \end{array}$$

A three-dimensional numerical avian model was developed to investigate the unsteady and turbulent aerodynamic performance of flapping wings for varying wingbeat frequencies and flow velocities (up to 12 Hz and 9 m s(-1)), corresponding to a reduced frequency range of k=0.22 to k=1.0 and a Reynolds number range of Re=16,000 to Re=50,000. The wings of the bird-inspired model consist of an elastic membrane. Simplifying the complicated locomotion kinematics to a sinusoidal wing rotation about two axes, the main features of dynamic avian flight were approximated. Numerical simulation techniques of fluid-structure interaction (FSI) providing a fully resolved flow field were applied to calculate the aerodynamic performance of the flapping elastic wings with the Reynolds averaged Navier-Stokes (RANS) approach. The results were used to characterize and describe the macroscopic flow configurations in terms of starting, stopping, trailing and bound vortices. For high reduced frequencies up to k=0.67 it was shown that the wake does not consist of individual vortex rings known as the discrete vortex ring gait. Rather, the wake is dominated by a chain of elliptical vortex rings on each wing. The structures are interlocked at the starting and stopping vortices, which are shed in pairs at the reversal points of the wingbeat cycle. For decreasing reduced frequency, the results indicate a transition to a continuous vortex gait. The upstroke becomes more aerodynamically active, leading to a consistent circulation of the bound vortex on the wing and a continuous spanwise shedding of small scale vortices. The formation of the vortices shed spanwise in pairs at the reversal points is reduced and the wake is dominated by the tip and root vortices, which form long drawn-out vortex structures.

In flows material does not vanish, nor does new material appear. The velocity fields therefore have to satisfy the law of conservation of mass. This law is easiest to formulate for steady flows if the shape of the streamlines is already known. We consider a stream filament through every cross-section of which the same amount of mass flows per unit time. If this mass were not the same in two cross-sections, the mass content of the stream filament between two crosssections would have to decrease or increase, contradicting the idea of a steady state. If A is the cross-section of the stream filament at a certain position, w the mean velocity in this cross-section, and ρ the associated density, then per unit time, the fluid volume $$A \cdot w$$ flows through the cross-section. The fluid mass flowing through the cross-section per unit time is $$\rho \cdot A \cdot w$$ . Continuity requires that $$\rho \cdot A \cdot w$$ must have the same value in all cross-sections of a stream filament. This implies that a stream filament of a steady flow cannot terminate in the interior of the fluid. It may extend from one boundary of the fluid space under consideration to the other boundary of the space, or it can turn back on itself.

We present a 3D code-coupling approach which has been specialized towards cardiovascular blood flow. For the first time, the prescribed geometry movement of the cardiovascular flow model KaHMo (Karlsruhe Heart Model) has been replaced by a myocardial composite model. Deformation is driven by fluid forces and myocardial response, i.e., both its contractile and constitutive behavior. Whereas the arbitrary Lagrangian–Eulerian formulation (ALE) of the Navier–Stokes equations is discretized by finite volumes (FVM), the solid mechanical finite elasticity equations are discretized by a finite element (FEM) approach. Taking advantage of specialized numerical solution strategies for non-matching fluid and solid domain meshes, an iterative data-exchange guarantees the interface equilibrium of the underlying governing equations. The focus of this work is on left-ventricular fluid–structure interaction based on patient-specific magnetic resonance imaging datasets. Multi-physical phenomena are described by temporal visualization and characteristic FSI numbers. The results gained show flow patterns that are in good agreement with previous observations. A deeper understanding of cavity deformation, blood flow, and their vital interaction can help to improve surgical treatment and clinical therapy planning.

Modeling the function of the human heart is of growing importance in a time when surgical treatment becomes the predominant therapy option. The hemodynamics of ventricular and vascular flow is closely linked to other disciplines like structural mechanics of myocardial and vascular tissue, electro-dynamical excitation, etc. A multi-disciplinary approach is therefore needed to describe it. While a monolithic solution of the underlying differential equations in one set is thinkable, a partitioned approach can take into account the specifics of the single disciplines and profit from specialized models and algorithms. We show how by coupling specialized solvers for fluid and solid mechanics a coupled model of ventricular flow can be created that gives insight into the hemodynamics in a way that is more than the sum of its parts.

A three dimensional mechanical and numerical avian model with identical geometry was developed to investigate the aerodynamic performance of flapping flight for varying flow velocities and wing beat frequencies. The corresponding reduced frequencies range from k=0.22 to k=1.0 providing turbulent and unsteady flow. The model consists of a rigid body and elastic wings. Its shape was inspired by birds, but restricted by manufacturing and numerical specifications. Using a sinusoidal flapping about an off-centre axis parallel to the body axis and a phase-shifted pitching about the moving lateral wing axis the wing beat motion was realized. Wind tunnel tests with Particle Image Velocimetry (PIV) were performed to capture the velocity field around and behind the mechanical model for different reduced frequencies. Furthermore, simulations for the corresponding numerical model have been conducted by means of fluid-structure-interaction (FSI) simulation techniques providing a fully resolved flow field. The results were used to analyze the flow configurations and to validate the numerical and experimental setup for further investigations. The results of the numerical simulations and wind tunnel experiments are in good agreement and facilitate a reconstruction of the three dimensional vortex structures in the wake. The results show, that for all reduced frequencies, the wakes consist of a chain of interlocked vortex rings behind each wing. For high reduced frequencies, a shedding of small-scale vortices composing vortex sheets generates oppositely rotating upstroke (UVS) and downstroke (DVS) vortex structures which contain starting, stopping, tip and root vortices. For decreasing reduced frequencies, the upstroke becomes more aerodynamically active leading to a diffusion of the upstroke vortex structures.

The Karlsruhe Heart Model (KaHMo) is a patient-specific simulation tool for a three-dimensional blood flow evaluation inside the human heart. Whereas KaHMo MRT is based on geometry movement identified from MRT data, KaHMo FSI allows the consideration of structural properties and the analysis of FSI. Previous investigations by Oertel et al. have shown the ability of KaHMo to gain insight into different intra-ventricular fluid mechanics of both healthy and diseased hearts. However, the in vivo validation of the highly dynamic cavity flow pattern has been a challenging task in recent years. As a first step, the focus of this study is on an artificial ventricular experiment, derived from real heart anatomy. Fluid domain deformation and intra-ventricular flow dynamics are enforced by an outer surface pressure distribution. The pure geometrical representation of KaHMo MRT can now be complemented by constitutive properties, pressure forces, and interaction effects using KaHMo FSI's partitioned code-coupling approach. For the first time, fluid domain deformation and intra-ventricular flow of KaHMo FSI has been compared with experimental data. With a good overall agreement, the proof of KaHMo's validity represents an important step from feasibility study toward patient-specific analysis.

Liquids are distinguished from solids by the fact that their particles are readily displaced. Whereas forces of finite magnitude are required to deform a solid, no force at all is required to alter the shape of a liquid, provided only that sufficient time is allowed for the change of shape to take place. When the shape is altered quickly, liquids do display a resistance, but this vanishes very quickly after the motion is finished. This ability of liquids to oppose a change in shape is called viscosity. We will discuss viscosity in depth in Section 4.2. As well as the usual liquids that are easy to move, there are also very viscous liquids whose resistance to change of shape is considerable, but which vanishes again at rest. Starting out from the viscous state, all phase transitions to (amorphous) solid bodies are possible. Heated glass, for example, passes through all possible transitions; in asphalt and similar substances these transitions occur at normal temperatures. For example, depending on the temperature, if a barrel of asphalt is tipped over, the asphalt will flow out within a few days or weeks. The mass that flows out forms a flat cake. Although it continually flows, one can walk on it without making footprints. Footprints will be left, however, if one stands still for a longer time on the asphalt. Hammering on the asphalt causes the mass to shatter like glass. In the study of the equilibrium of liquids, we consider states of rest or sufficiently slow motion. The resistance to change of shape may then be set to zero, and we obtain a definition of the liquid state: In a liquid in equilibrium, all resistance to change of shape is equal to zero.

When a viscous fluid flows through long straight tubes at reasonably high speeds, the Hagen–Poiseuille law (4.45), according to which the pressure drop is linearly proportional to the volume of fluid flowing through the pipe, is replaced by another law, in which the pressure drop is significantly greater, and almost proportional to the square of the volume flow rate of fluid. At the same time it is found that the flow field, which is smooth and straight (or laminar) in the Hagen–Poiseuille regime, becomes at higher velocities full of irregular eddying motions (or turbulent). This may be seen clearly in the case of a fluid flowing through glass tubes if a dye is introduced through a small injector at the inlet (Figures 6.1, 4.52). The colored filament is straight and smooth for low speeds but breaks off and disperses almost uniformly when turbulence develops. As a second example, introduced in Chapter 1, consider a jet of water that emerges from a circular orifice into a tank of still water. At very low speeds of the fluid the jet is smooth and steady. For higher speeds, it develops swirls of various sizes amidst avalanches of complexity.

The flow rate measurement of liquid, steam, and gas is one of the most important areas of application for today’s field instrumentation. Vortex meters are used in numerous branches of industry to measure the volumetric flow by exploiting the unsteady vortex flow behind a blunt body.

In contrast to the topics discussed in previous chapters, biofluid mechanics is concerned with flows that are influenced by flexible biological surfaces. We distinguish between flows past living bodies in air or in water, such as bird flight or the swimming of fish, and internal flows, such as the closed blood circulation of living beings. In the previous millions of years, evolution has developed crawling, running, swimming, gliding, and flying as methods of motion of living beings, depending on their size and weight. The necessary propulsion for altering position requires flow control adapted to the Reynolds number of the motion. The motion of bacteria and amoebae takes place at very small Reynolds numbers, where friction dominates, with cilia and flagella. Tadpoles and octopuses use the inertia of jet propulsion for motion. Eels move in a wavelike manner; whales use vortex separation at the tail fin for motion at Reynolds numbers of up to 108. Fast-swimming fish such as the shark have longitudinal grooves on their scales that affect the viscous sublayer of the flow boundary layer so that the flow drag is reduced. Heat and mass transport in living beings takes place in circulatory systems. These include respiration, the circulatory system of the blood, and the lymph, as well as that of water. All biological flows have in common that the motion is affected by external or internal highly flexible and structured surfaces. This leads to an actively controlled flow whose losses are kept small. Of the many different biological flows, in this chapter we will treat the flying of birds, the swimming of fishes as well as looking at the blood circulation of the human body in depth.

The flows of liquids and gases have so much in common that it is practical to treat them together. In contrast to liquids, gases are compressible. However, whether the compressibility is important depends on the flow process under consideration. At small velocities and for moderate height dimensions of the gas, the pressure changes remain small compared to the mean pressure. The volume changes are then so small that they can be neglected. Gas flows are then no different from flows of incompressible liquids. If we neglect volume changes of 1%, we can apply the equations for incompressible flows to flows in the atmosphere at mean temperatures. This remains the case for velocities of up to 50 m/s and for height dimensions of up to 100 m (cf. Sections 2.5 and 4.1.2). At flow velocities of 150 m/s, the volume changes are about 10%. If the flow velocities reach the magnitude of the velocity of sound (about 340 m/s), the volume changes become so large that the flow is greatly affected by them. At flow velocities that are greater than the velocity of sound, the flow has a completely different character from that of an incompressible liquid. In this chapter we mainly consider incompressible flows. In order not to have to speak of liquids and gases, we use the word fluid as a collective term for liquids and gases. For the purposes of this usage, gases are referred to as compressible fluids (Section 4.3). The kinematics of a flow describes the motion of the fluid without taking into account the forces that cause this motion. The goal of kinematics is to describe the dependence of the motion of the fluid elements on time for a given velocity field.

This chapter on convective heat and mass transfer starts out from Prandtl’s original chapter Heat Transfer in Flowing Liquids. We will treat free convection flows, caused by the density changes in the fluid due to temperature and concentration gradients. These cause a lift in the gravitational field, which in turn causes convection flows. Examples of free convection flows at heated cylinders and plates were shown in the introductory chapter in Figure 1.6. The Rayleigh–Bénard convection of Figure 1.5 and diffusion convection are also examples of free convection flows. Forced convection flows occur when an external force, such as a pressure gradient, also acts on the flow. Forced convection flows occur, for example, in heated or cooled pipes such as those used in heat exchangers. Heat and mass exchange processes are found in the ocean and in many different processes in chemical technology, such as absorption, adsorption, extraction, and distillation. When water evaporates on the surface of the oceans, a high salt concentration remains, and an unstable density layer with diffusion instabilities occurs. The expansion of substances in solvents and the separation of substances in centrifuges are further examples. Examples of biological mass exchange processes are the supply of oxygen to the blood and absorption of food in the body.

Introduction.- Properties of Liquids and Gases.- Kinematics of Liquids and Gases.- Dynamics of Liquids and Gases.- Fundamental Equations of Fluid Mechanics.- Aerodynamics.- Turbulent Flows.- Fluid Mechanical Instabilities.- Convective Heat and Mass Transport.- Multi-Phase Flows.- Flows with Chemical Reactions.- Flows in the Atmosphere and in the Ocean.- Biofluid Mechanics.- Thermal Flow Machinery.

The flows in Earth's atmosphere (air flows) and in the oceans (oceanic currents) do not differ in principle from those flows in technical areas treated in the previous chapters. These are motions of gases (atmosphere) and liquids (ocean) that are acted on by gravity and are determined by pressure and frictional forces. The atmosphere and the oceans are part of a rotating system in which Coriolis and centrifugal forces also act. From this point of view, a common treatment of flows in the atmosphere and in the ocean seems sensible. In fact, the name geophysical fluid dynamics has come to describe these flows. In particular, we refer to the textbooks of B. Cushman-Roisin (1994), A. E. Gill (1982), and J. Pedlosky (1994). Indeed, Ludwig Prandtl, in the original Essentials of Fluid Mechanics, considered flows in the atmosphere and oceans at several points throughout the book. In this chapter we point out the essential elements of the geophysical flow processes in the atmosphere and oceans as discussed by Prandtl. Naturally, we can present only some elements of this topic, and this chapter in no way replaces the specialized literature from the areas of meteorology and oceanography.

BACKGROUND:The efficacy of surgical ventricular reconstruction (SVR) for ischemic cardiomyopathy has never been truly quantified. Methods to assess ventricular flow have not been applied to these patients. The objective is to develop a volume-independent technique for assessing the effects of ischemic remodeling and SVR on left ventricular blood flow dynamics. METHODS:Cardiac magnetic resonance images from a healthy volunteer and from a patient before and after SVR were segmented and transformed to generate a grid model of the heart by generating numeric grids and running third-order approximations to achieve 850 grid images per cardiac cycle. These grids formed the skeletal structure of our patient-specific time-dependent ventricular geometry model, the Karlsruhe Heart Model, used for modeling fluid dynamics. We modeled flow, ejection fraction, and blood washout from the ventricle. The model was validated using a silicone ventricle and mock circulation. RESULTS:In the healthy heart and before SVR, ejection fractions were 0.61 and 0.15 and left ventricular volumes were 166 mL and 175 mL, respectively. Surgical ventricular reconstruction decreased left ventricular volume by one fourth. Postoperative ejection fraction was 0.18 in the patient. Post-SVR shape was more spherical than preoperatively and also more spherical than the healthy heart. Ventricular flow patterns in the patient were significantly altered by SVR. However, fluid washout from the ventricle was similar before and after SVR but worse than in the healthy heart. CONCLUSIONS:Fluid dynamic modeling of the heart is possible based on cardiac magnetic resonance imaging data and enables volume-independent quantitative assessment of the surgical procedure. In the future, preoperative modeling for patients with remodeled ventricles may help to achieve optimized post-SVR flow characteristics and potentially outcomes.