Springer Handbook of Experimental Fluid Mechanics

A fluid flow experiment is an attempt to isolate a part of the world and measure flow and thermodynamic properties. A fluid is defined as a material that deforms continuously if a shear stress is applied. An internal flow situation has walls bounding the flow, but an inflow and outflow position must be controlled. An external flow problem has a uniform flow far from the body of interest. In both situations the state of flow at the boundary is controlled. In the mathematical representation of the flow, the flow conditions on the boundary are specified. This is the nature of the governing physics. If the boundary conditions depend on time the flow situation in the entire region must be specified at the initial time.

In what follows the major physical laws are outlined. In most cases tensor calculus in symbolic form is employed. Scalars are lightface type, vectors are boldface type, and tensors are boldface capitals. However, in cases where confusion is possible with tensor multiplications, index notation is employed. Scalars are then without an index, vectors have one index and tensors have two or more indices.

Given that an experiment can be considered to represent a physically realizable boundary value (bv) problem and given that the derived measurements are to represent aspects of the solution to the bv problem, it is rational to extend this understanding such that a maximum amount of information can be obtained from a given experiment. The first portion Sect. 2.2.1 establishes the bases for obtaining information regarding the flow associated with a prototype (the object/flow of actual interest) from measurements made in a model study. This section focuses on the large class of flows for which a Newtonian fluid and its governing equations establish the model-to-prototype information exchange.

Dimensional analysis Sect. 2.2.2 provides a complement to Section 2.1 with a less structured - and therefore a more flexible - approach to problems that extend beyond those readily addressed by the Sect. 2.2.1 material. The important issue of collecting experimental results in non-dimensional groups is addressed in Sect. 2.2.2.

The discussion of self-similarity Sect. 2.2.3 addresses the immense compaction of experimental data that is made possible for those flows that exhibit this property. The bases for, and utilization of, self-similarity are explored in detail.

The density of a fluid is defined as the mass of the fluid per unit volume. Some methods of measuring the density of the fluid with high precision determine the variation of one quantity in this ratio when the other is fixed and a further state variable is altered. Other methods make use of the effect of the fluid density on the position or motion of a rigid body contained within it. Examples of these instruments are described in this chapter for operation over a wide range of thermodynamic states.

Measurements of the steady pressure in a fluid flow may be required to determine other thermodynamic properties, to determine forces on a body due to the pressure distribution over it, or in order to determine the dynamic head and flow velocity (for further details on the latter see Sect. 5.1. Pressure is a scalar representation of molecular activity, a measure of the nondirectional molecular motions. Thus it must, by definition, be measured by a device at rest relative to the flow. Whilst the common practice in the fluid mechanics community is to denote the pressure as static (as opposed to the coordinate-dependent total pressure, Sect. 3.1), this terminology introduces a fundamental redundancy.

In practice, pressure is commonly measured both at walls and in the freestream using the types of measurement device shown in Fig. 4.1 connected to a transducer of suitable sensitivity and range. The orifice of a small wall tapping represents a simple way to obtain the pressure impressed on the wall by the external flow. So-called static pressure tubes approximate the local fluid pressure in the freestream if the disturbance presented to the flow can either be accounted for or is not large to begin with. However this can only ever be strictly true for steady laminar flow due to the normal velocity component introduced when a flow becomes turbulent. Measurement of freestream pressure is one of the hardest challenges in fluid mechanics.

This chapter addresses measurement of pressure using wall tappings (Sect. 4.1) and static pressure tubes (Sect. 4.2), and especially errors due to the intrusive flow presence of real, finite-sized devices and calibrations to correct for these. Bryer and Pankhurst [4.1] and Chue [4.2] provided seminal monographs on the general topic of pressure probes in 1971 and 1975, respectively, which give detailed descriptions of measurement devices, coverage of the background to the various corrections and a survey of older data. The topic is covered here more concisely, with a view to

practical use by the engineer, and with reference to modern literature. The reader is referred to Bryer and Pankhurst [4.1] and Chue [4.2] for further details on most sections.

In more recent years a further method for obtaining pressure on the surface of a wind tunnel model has been developed, based on pressure sensitive paints (PSP). The introduction of PSP provides a method to measure the pressure on the surface of a model directly without the transducers and tubing associated with conventional means. A paint, the luminescence of which is dependent on air pressure, is applied to the surface of a wind tunnel model and the pressure distribution is obtained from the images produced by proper illumination. In Sect. 4.4 the basics of PSP are discussed and further subsections address in detail different paints, paint application procedures, imaging systems and image processing. In discussing the achievable accuracy of PSP techniques, both the spatial and temporal resolution is examined. The thermal sensitivity of the paint dye is introduced and this is closely linked to temperature-sensitive paints (TSP), as discussed in Chap. 7, Sect. 7.4.

The objective of this chapter is to provide a comprehensive statement of the experimental methods that can be used to transduce the velocity and its companion quantity: vorticity (∇ × u ¯). Velocity measurements can be understood to represent spatially integrated and pointwise values. Thermal transient anemometry (Sect. 5.6) and sonic anemometers (Sect. 5.7) represent the former. Pressure-based velocity measurements (Sect. 5.1), thermal anemometry (Sect. 5.2), and particle-based techniques (Sect. 5.3) represent the latter. In addition, particle image velocimetry (PIV, Sect. 5.3.2), planar Doppler velocimetry (Sect. 5.3.3), and molecular tagging velocimetry (Sect. 5.4) also provide spatial distributions of the pointwise measurements for the instant at which the image is formed. The vorticity measurements rely on some form of the above pointwise measurements. A general overview of optical methods is presented in Sect. 5.5.1.

The methods dealt with in this section are based on changes of fluid density; hence its index of refraction. As a result of these changes, optical phase and, coupled with it, direction of propagation of a light wave transmitted through the flow are altered in comparison to the properties of the incident light. The available signal can be presented in planar form, i.e., as a flow picture, and the methods are often referred to as optical flow visualisation, because the changes in index of refraction are detected and measured by optical techniques. The obtainable information is integrated along the whole path of the light in the fluid field (line-of-sight methods) and, in a three-dimensional (3D) object field, special techniques for interpreting the signal pattern are necessary (tomography) in order to provide local data values of the quantity to be determined, e.g., density. Four major groups of experimental methods can be distinguished: shadowgraphy, schlieren technique, moiré techniques, and interferometry. The fluid mechanical problem areas to which these optical measuring techniques can be applied are compressible flow, convective heat transfer, mixing and mass transfer, combustion, and flows with density stratification.

Thermochromic liquid crystals (TLCs) can be applied for thermographic measurements of heat transfer and temperature in fluid mechanics, delivering important quantitative full-field data for comparison with and validation of numerical simulations. Thin coatings of TLCs at surfaces are utilized to obtain detailed heat transfer data for steady or transient processes. Application of TLC tracers allows instantaneous measurement of the temperature and velocity fields for two-dimensional cross sections of flows. These methods are based on computerized true-color analysis of digital images for temperature measurements and modified particle image velocimetry, which is used to obtain the flow field velocity. In this Chapter, the advantages and limitations of liquid-crystal thermography are discussed, followed by several examples of thermal flow field measurements.

The use of infrared thermography for non-intrusive measurement of spatially resolved surface heat transfer characteristics is described for five different measurement environments, including situations where large gradients of surface temperature are present. In the first of these, measurements are made on the surface of a therapeutic biomedical patch, where the quantity of interest is the time-varying spatially resolved surface temperature. For the other situations, the measured temperature distributions are used to deduce quantities such as the surface Nusselt numbers on the surface of a swirl chamber, the effectiveness of surface adiabatic film cooling downstream of individual shaped film cooling holes, the surface heat flux reduction ratio downstream of two rows of film cooling holes placed on a model of the leading edge of an airfoil, and thermal boundary condition information for numerical predictions of the heat transfer characteristics on the surface of a passage with an array of rib turbulators. In all of these situations, in situ calibration procedures are employed in which the camera, imaging, and data-acquisition systems are all calibrated together in place within the experimental facility as the infrared measurements are obtained. This requires separate, simultaneous, and independent measurements of surface temperatures, and produces spatially resolved results from infrared images with high levels of accuracy and resolution.

Measurement of steady and fluctuating forces acting on a body in a flow is one of the main tasks in wind tunnel experiments. In aerodynamic testing, strain gauge balances will usually be applied for this task as, particularly in the past, the main focus was directed on the measurement of steady forces. In many applications, however, balances based on piezoelectric multicomponent force transducers are a recommended alternative solution. Contrary to conventional strain gauge balances, a piezo balance features high rigidity and low interference between the individual force components. High rigidity leads to very high natural frequencies of the balance itself, which is a prerequisite for applications in unsteady aerodynamics, particularly in aeroelasticity. Moreover for measurement of extremely small fluctuations, the possibility exists to exploit the full resolution independently from the preload.

Concerning the measurement of small, steady forces, the application of piezo balances is restricted due to a drift of the signal at constant load. However, this problem is not as critical as generally believed since simple corrections are possible.

The aim of this chapter is to give an impression of the possibilities, advantages and limitations offered by the use of piezoelectric balances. Several types of external balances are discussed for wall-mounted models, which can be suspended one-sided or twin-sided. Additionally an internal sting balance is described, which is usually applied inside the model. Reports are given on selected measurements performed in very different wind tunnels, ranging from low-speed to transonic, from short- to continuous running time and encompassing cryogenic and high pressure principles. The latter indicates that special versions of our piezo balances were applied down to temperatures of −150 °C and at pressures of up to 100 bar.

The projects span from a wing/engine combination in a low-speed wind tunnel to flutter tests with a swept-wing performed in a transonic wind tunnel, and include bluff bodies in a high pressure and cryogenic wind tunnel, as well. These tests serve as examples for discussing the fundamental aspects that are essential in developing and applying piezo balances. The principle differences between strain gauge balances and piezo balances will also be discussed.

Rheological constitutive equations of non-Newtonian liquids are discussed in detail in Chap. 1.8. In the present chapter they are used in the discussion of measurement techniques intended to establish an appropriate constitutive equation of a given liquid and attribute values for its material parameters. As is widely done in rheology, shear viscosity (not necessarily constant) is always denoted by η; only for Newtonian liquids, where η = constant, is it denoted by μ as elsewhere in this Handbook.

Fluid flows in nature and technology normally depart from laminarity and are turbulent in the majority of cases, including flows around bodies such as airplanes, vehicles, ships, and in internal flows such as in ducts, turbomachines, propulsors, and even in blood circulation in the human body. Laminarity is the anomaly and not the standard. As will be shown in this chapter, the parameter which is fundamental to the transition from laminarity to turbulence is the Reynolds number, i.e., the ratio of inertial to viscous forces. In Sect. 10.1 the statistical Eulerian description of turbulent flows will be developed followed by a section on Reynolds decomposition and Reynolds equations. Section 10.1.3 finally surveys scales in turbulent flows.

In Sect. 10.2 the optical Lagrangian particle-tracking technique, capable of producing robust, single- and multiparticle Lagrangian measurements, is presented. First the image-processing algorithms used to determine the particle trajectories are discussed and then the implementation of the technique in the laboratory is described. A brief presentation of results focusing on the separation of particle pairs in intense turbulence is also given.

In Sect. 10.3 a novel type of random flow in a dilute polymer solution of a flexible high-molecular-weight polymer in two different flow setups that share the same feature of high curvature of the flow lines is discussed. In the first part of this section the hydrodynamic description of dilute polymer solution flows and the nondimensional parameters that follow from these equations to characterize these flows are presented. Variation of one of these control parameter responsible for the elastic properties of a fluid can lead to a new elastic instability in various flows that is distinguished by the presence of curvilinear trajectories. The theoretical criteria for this elastic instability in three different flows together with experimental verification are discussed. To complete the basics, the rheometric properties of the polymer solutions used and their relation to Boger fluids are given. The first observation of elastic turbulence, in the flow between two plates, is described. Then the experimental measuring techniques used to characterize the flow are given, and a complete description of the results of measurements together with a discussion of the results is presented. Finally, the role of elastic stress, a recent theory of elastic turbulence, and comparative studies of elastic versus hydrodynamic turbulence are discussed. The last part of the section deals with the description of the elastic turbulence in a curvilinear channel or Dean flow, where a particularly detailed experiment on mixing due to elastic turbulence was conducted. A summary of the results is given finally.

Section 10.4 briefly reviews large-eddy simulations (LES) and the specific data requirements for LES (Sect. 10.4.1) and then describes the experimental methods that have been employed to obtain such data starting with arrays of point-measurement techniques (Sect. 10.4.2) and optical planar velocimetry measurement methods (Sect. 10.4.3). Sample results from the latter applied to studies of LES models are presented in (Sect. 10.4.4). The application of optical volumetric techniques for three-dimensional (3-D) velocity measurements are described in Sect. 10.4.5. Scalar fluctuation measurements using optical techniques and their applications to the study of LES variables of interest to scalar mixing and combustion are reviewed in Sect. 10.4.6.

Turbulent wall-bounded flows (i.e., boundary layer, pipe and channel flows) present additional measurement challenges relative to those in, say, free shear turbulent flows or grid turbulence. The physical presence of the wall and the limitations and influences it presents on the implementation of sensing technologies creates some of these challenges. Other, often more-subtle issues, however, relate to the effect that the wall has on the inherent flow dynamics. Such effects are reflected in the steep mean velocity gradient(s) in the vicinity of the surface, as well as the length and time scales of the turbulence local to the near-wall region. Regarding the latter, primary challenges are associated with the high frequencies and small scales of near-wall turbulence relative to free shear flows.

In previous Chaps. (5.2, 5.3 and 5.5.3), relatively broad discussions were provided regarding the requirements and considerations for accurate measurements of both mean and fluctuating quantities in turbulent flows. The present chapter constitutes an extension of these more-generic considerations relative to the specific case of wall-bounded turbulent flows. For the purposes of providing a background context, the initial subsection below presents a brief overview of concepts and considerations specific to wall flows. Owing to its central role in the study of the turbulent wall flows, the next subsection addresses the measurement of the wall shear stress for ca nonical boundary layer, pipe and channel flows. Considerations relative to transitional and non-canonical wall flows are presented in subsequent subsections.

The vector field variables of interest in experimental fluid mechanics include velocity, vorticity and wall shear stress. Additional vector fields of interest include the gradients of pressure and temperature. Each of these is subject to the constraints that can be placed on the isolated singular points (vector magnitude = 0) of the vector field that is projected onto a surface of interest. Identifying the relevant surfaces for a given flow and establishing, for a given identified surface, an a priori constraint on the isolated singular points of the relevant vector field, can provide experimentalists with a powerful diagnostic tool.

It is the purpose of this chapter of the handbook to enable the reader to utilize this diagnostic tool for any flow field of interest.

This chapter focuses on measurement techniques that have been used during experimental investigations of turbomachinery flow fields. These techniques are not fundamentally different from those used in other external flow studies. However, implementing them within turbomachines has introduced a series of unique and specialized issues in the preparation of the experimental setup, data acquisition, and analysis procedures. This chapter provides detailed information on the methods used to address these issues, along with a comprehensive summary on how they have been implemented to investigate complex flow phenomena within turbomachinery components.

The three segments of this chapter introduce phenomena that are of specific interest in the area of hydraulics. Where applicable (Cavitation, Sect. 15.1 and Sediment Transport, Sect. 15.3) introductory and descriptive material regarding the topic is provided. Terminology, physical examples and motivating descriptions introduce the comprehensive Cavitation subsection. Examples of the types of flows in which cavitation occurs are provided. This information sets the stage for a description of the types of facilities and instrumentation that are necessary to study the problem. Numerous photographs and descriptive sketches clarify and complement the text.

The wave height measurement segment first deals with fixed position single "point" techniques. More advanced techniques for: i) wave surface shape along a horizontal line, and ii) two-dimensional surface geometry measurements for laboratory and field observations are then described.

Following the introduction to sediment transport phenomena and terminology, the methods of measurement: manual, optical and acoustic are given detailed descriptions including calibration techniques for the latter two methods. Bed load sediment measurements: pressure difference, sediment trapping and acoustic are next described. Total load and the less common measurement techniques, plus references complete the subsection.

Vehicle aerodynamics has its roots in aviation, and the same is true for the related experimental techniques. The wind tunnel is the key carry-over element. Wind tunnels have been adapted stepwise to meet the specific needs of vehicle aerodynamic studies. Three areas of development must be mentioned: With vehicles a much greater blockage can be tolerated; to simulate the relative motion between vehicle and road or track different belt systems and mechanisms to reproduce wheel rotation have turned out to be useful; for tests in various climates a variety of specialized smaller wind tunnels have been developed. Measurement techniques are very much the same as in aeronautics at low Mach number and increasingly particle image velocimetry (PIV) is being applied to study the complex flow fields around vehicles. Nevertheless, flow visualization and photogrammetry still remains an indispensable tool for a quick evaluation of different geometries, also for dynamic flow conditions.

A selected group of common instruments used to measure the atmosphere is described in this chapter. Typical atmospheric measurements include winds, temperature, pressure, humidity, dew point, moisture, radiation, visibility, cloud heights, lightning, gaseous composition, aerosols, and precipitation. Atmospheric sensors may measure at a point (in space) or remotely (with a given distance from the volume of air being measured). As an example of a common wind sensor, sonic anemometers measure wind speed in two or three directions using differences in the propagation speed of sound in different directions across moving air. The surface heat budget, e.g., solar radiation, albedo, the Earthʼs long-wave radiation, heat fluxes into the soil, and latent/sensible heat fluxes are major forcing effects for atmospheric motions. Standard measurement methodologies are presented in Sect. 17.1.3. Some of the most frequently used measuring instruments for atmospheric dispersion are reviewed in Sect. 17.2. Major field experiments have recently used arrays of both bag samplers and fast response sensors to track the dispersion of plumes of trace gases in urban areas. Remote sensing instruments, both active and passive, are becoming increasingly available and robust. Remote sensing instruments may be ground-based or satellite based. While the flexibility and power of modern remote sensing instruments is impressive, there is frequently an inversion challenge associated with data interpretation.

This chapter describes various methods to measure the oceanographic variables that are dynamically significant. After a brief overview of the various motions and their temporal and spatial scales, the challenges of making measurements in a vast, inhospitable, and unforgiving environment are described. Then point measurements (pressure, temperature, salinity, sound speed, density, and velocity), Lagrangian measurements (floats and dye dispersion), and remote sensing methods (acoustic and electromagnetic) are described. Because many of the practical problems of oceanographic measurements are associated with the complete measurement system, examples of these and illustrative case studies of several experiments are given.

The no-slip boundary condition at a solid–liquid interface is at the center of our understanding of fluid mechanics. However, this condition is an assumption that cannot be derived from first principles and could, in theory, be violated. In this chapter, we present a review of recent experimental, numerical and theoretical investigations on the subject. The physical picture that emerges is that of a complex behavior at a liquid/solid interface, involving an interplay of many physicochemical parameters, including wetting, shear rate, pressure, surface charge, surface roughness, impurities and dissolved gas.

In Sect. 19.1 we present a brief history of the no-slip boundary condition for Newtonian fluids, introduce some terminology, and discuss cases where the phenomenon of slip (more appropriately, this may often be apparent slip) has been observed. In Sect. 19.2 we present the different experimental methods that have been used to probe slip in Newtonian liquids and summarize their results in the form of tables. A short presentation of the principle and results of molecular dynamics simulations is provided in Sect. 19.3, as well as remarks about the relation between simulations and experiments. We then present in Sect. 19.4 an interpretation of experimental and simulation results in light of both molecular and continuum models, organized according to the parameters upon which slip has been found to depend. We conclude in Sect. 19.5 by offering a brief perspective on the subject.

Combustion processes consist of a complex multidimensional interaction between fluid mechanics and chemical kinetics. A comprehensive experimental analysis needs therefore measurements of flow and scalar fields. These measurements need to be performed in-situ with high temporal and spatial resolution as well as high accuracy and precision. In addition, any disturbances during the measurement should be avoided. These requirements are fulfilled best by laser-optical techniques. Whereas flow fields are commonly measured by methods like laser Doppler or particle imaging velocimetry discussed elsewhere, the focus of this chapter is on scalar measurements based on spectroscopy. Scalars of interest are temperatures, chemical species concentrations, or rate of mixing between fuel and oxidant. Following an introduction, Sect. 20.3 presents the interconnection between experimental analysis and numerical simulation of combustion processes. In Sect. 20.4, various spectroscopic techniques are described exemplary in their application to different fields of combustion research. The chapter concludes with aspects on future developments in combustion diagnostics.

In this chapter we review briefly the fundamentals of electrohydrodynamics (EHD), the characteristic EHD dimensionless numbers and the techniques to measure conductivity and electric field, as well as the peculiarities imposed by charging of particles in the classical fluid-mechanical methods for measuring velocity and visualizing fluid flows.

We begin with a brief review of the basic equations, followed by an examination of the physical mechanisms that govern fluid flow through the relevant dimensionless numbers related to electric forces. However, the main emphasis is put on the description of the experimental methods, used to measure the fundamental EHD magnitudes. First, we discuss the basic mechanisms of conductivity, how to measure it, and how to obtain reproducible I–V characteristics. This section also includes a discussion of the techniques to control ion injection. This is followed by a section dedicated to the measurement of mobility. Then we describe the Kerr effect, and how it can be used to measure the electric field in liquids. The last section is dedicated to a description of the difficulties we encounter in the classical techniques of laser Doppler anemometry, and visualization techniques in EHD flows, and how they may be overcome, at least partially. We hope that this chapter will be useful, not only to EHD researchers, but also to practising fluid dynamicists, and to chemical and electrical engineers who need to understand and apply the principles and experimental techniques of EHD in their work.

This chapter is devoted to reviewing some fundamental transforms and analysis procedures commonly used for both signal and data processing in fluid mechanics measurements. The chapter begins with a brief review of the Fourier transform and its digital counterpart the discrete Fourier transform. In particular its use for estimating power spectral density is discussed in detail. This is followed by an introduction of the correlation function and its relation to the Fourier transform. The Hilbert transform completes the introductory topics. The chapter then turns to a rigorous presentation of the proper orthogonal decomposition (POD) in the context of the approximation theory and as an application of singular value decomposition (SVD). The relationship between POD and SVD is discussed and POD is described in a statistical setting using an averaging operation for use with turbulent flows. The different POD approaches are briefly introduced, whereby the main differences between the classical POD and the snapshot POD are highlighted. This section closes with a presentation of the POD as a generalization of the classical Fourier analysis to inhomogeneous directions. The chapter continues with a discussion of conditional averages and stochastic estimation as a means of studying coherent structures in turbulent flows before moving in a final section to a comprehensive discussion of wavelets as a combination of data processing in time and frequency domain. After first introducing the continuous wavelet transform and orthogonal wavelet transform their application in experimental fluid mechanics is illustrated through numerous examples.

In this chapter the fundamentals of statistical parameter estimation are reviewed for applications typical in experimental fluid mechanics. The chapter begins with a review of the probability density function and its moments and continues with common estimators for the mean and variance of stationary random processes. A brief introduction to signal noise is given as a prelude to a rigorous discussion of the Cramér–Rao Lower Bound (CRLB). The CRLB represents the lower bound of variance of unbiased estimators of a parameter. This concept is deepened using illustrations from the laser Doppler, phase Doppler and PIV measurement techniques. The chapter closes with a short discussion about the propagation of errors in a measurement chain.

Imaging sensors convert radiative energy into an electrical signal and such sensors are available that cover the wide spectrum from gamma rays to the infrared. They accumulate an electrical signal during the exposure time and convert all the signals of an array of detectors into a time-serial analog or digital data stream. The dominate and most successful devices to perform this task are charge coupled devices (CCD). However directly addressable imaging sensors on the basis of CMOS fabrication technology are becoming more and more promising because the image acquisition, digitalization and preprocessing can be integrated on a single chip; hence yielding very fast frame rates. This chapter provides a comprehensive survey of the available imaging sensors, details the parameters that control their performance and gives practical tips to select the best camera for different imaging tasks.

From the beginning of science, visual observation has played a major role. At that time, the only way to document the results of an experiment was by verbal description and manual drawings. The next major step was the invention of photography more than one and a half centuries ago, which enabled experimental results to be documented objectively. In experimental fluid mechanics, flow visualization techniques gave direct insight into complex flows, but it was very difficult and time consuming to extract quantitative measurements from photographs and films.

Nowadays, we are in the middle of a second revolution sparked by the rapid progress in both photonics and computer technology. Sensitive solid-state cameras are available that acquire digital image data, and standard personal computers and workstations have become powerful enough to process these data. These technologies are now available to any scientist or engineer. As a consequence, image processing has expanded and continues to expand rapidly from a few specialized applications into a standard scientific tool.

This chapter gives a brief presentation of some of the most important general image processing techniques that are required to process image data in experimental fluid mechanics. The second section (Sect. 25.2) deals with motion analysis. The most important methods are introduced and classified according to the fundamental principles, assumptions and approximations upon which they are based.