1 Introduction

Dimensional analysis offers methods for reducing complex physical problems to the simplest form prior to obtaining a quantitative answer. At the heart of dimensional analysis is the concept of similarity. Mathematically, similarity refers to a transformation of variables that leads to a reduction in the number of independent variables that specify the problem.

Dimensional analysis and theory of similarity are in common use in model investigations/tests of different issues of many important physical phenomena not properly recognized up to now. In such cases, fulfilment of the respective similarity criteria becomes indispensable. For example, Barenblatt [2] considered problems of scaling, self-similarity and intermediate asymptotes; Flaga A. [9] considered basic principles and theorems of dimensional analysis and theory of model similarity of physical phenomena; Siedow [26] described and analysed similarity and dimensional analysis in mechanics; Sonin [29] described the physical bases of dimensional analysis; Ziereb [32] carried out problems of similarity criteria and modelling in fluid mechanics.

A very practical sourcebook on similarity methods and modeling techniques were elaborated by Westine et al. [31]. The problems involve spread of oil slicks, explosive cratering, car crashes, space vehicle heat exchange, explosive forming, and more, including alternate methods for developing model laws.

Many applications of dimensional analysis and the theory of similarity of physical phenomena occurring in various issues of engineering, can be found in the books, monographs and publications devoted to:

  • Wind engineering and aerodynamics of buildings and structures i.e.,

  • Similarity numbers important in problems of wind action on building structures encountered in their design (Cook [4]),

  • Original similarity criteria elaborated and analyzed by Flaga A. and Flaga et al. for different special issues, e.g., wind vortex-induced excitation and vibration of slender structures [6]; resistant of freight railway vehicles to roll-over in strong winds [22]; linear building objects at aerodynamic and gravitational actions [8]; wind tunnel model tests of two free-standing lighting protection masts [11]; sectional model of power line free-cable bundled conductors at their aeroelastic vibrations [12]; relation between shape and phenomenon of flutter of bridge deck-line bluff bodies [25],

  • Similarity criteria in aeroelastic model tests of flutter phenomenon for cable-stayed bridges (Flamand et al. [17]) and suspension bridges (Simiu and Scanlan [27]);

  • Environmental engineering, i.e.,

  • Environmental effects on buildings, structures and people: actions, influences, interactions, discomfort (Flaga et al. [10]),

  • Similarity criteria and problems of their fulfilment in different issues of environmental engineering (Flamand [16]);

  • Wind turbines, i.e.,

  • Original similarity criteria for authorial models of different types of vertical axis and horizontal axis wind rotors (Flaga [7]),

  • Modelling and the performance of different wind turbines, e.g. diffuser augmented wind turbine, stepped blade wind turbine, vertical axis wind turbine (Flay [18]), cross flow wind turbine above windbreak fence (Nakata et al. [24]),

  • Modelling and experimental investigations of the performance of a cross-flow wind turbine with and without diffuser (Akhgari et al. [1]);

  • Hydro-mechanics, hydraulics and fluid dynamics, i.e.,

  • Similarity, dimensional analysis and critical numbers at different fluid–solid relative motions (Blevins [3].),

  • Similarity criteria for fluid flows in conduits, fluid flows in channels, floating objects, in turbomachinery (Dougherty and Franzini [5]);

  • Snow engineering, i.e.,

  • Snow load distributions on different stadium roofs (Flaga and Flaga [13]),

  • Aerodynamic and aeroelastic similarity criteria for wind tunnel model tests of overhead power lines in triangular configuration under different icing conditions (Flaga et al. [14]),

  • Similarity criteria for wind tunnel model tests of snow precipitation and snow redistribution on rooftops (Kimbar and Flaga [21]), terraces and in the vicinity of high rise buildings (Flaga et al. [15],);

  • Different issues of structural mechanics, engineering dynamics, thermomechanics, aircraft flowing qualities, i.e.,

  • Hull girder ultimate strength assessment based on experimental results and the dimensional theory which maintains the first-order similarity between the model and real structures (Garbatov et al. [19]),

  • Design of scaled down model for structural vibration analysis of a tower crane mast using similitude theory as well as numerical modal analyses for prototype and two scaled models (Kenan et al. [20]),

  • Dimensional and similitude analysis of stiffened panels under longitudinal compression considering buckling behaviours (Song et al. [28]),

  • Similarity criteria for thin-walled cylinders subjected to coupled thermo-mechanical loads including thermo-elastoplastic failure behaviors predicted by numerical model validated by experiments (Ma et al. [23]),

  • Flying qualities criteria for scaled-model aircraft based on similarity theory taking into account relations between configuration parameters, control law parameters, and flight condition parameters (Wang et al. [30]).

From the above review of literature it can be stated that there are many different ways of formulating and defining dynamics similarity criteria in engineering.

A dimensional analysis can be carried out e.g. in relation to general functional relationships describing the phenomenon. After carrying out dimensional analysis on these functional relationships, in obtained dimensionless functional relationships appear set of nondimensional numbers being monomials created from main dimension or/and dimensional quantities (i.e. variables and parameters) characterizing this phenomenon which constitute the respective similarity criteria.

A dimensional analysis can also be carried out by considering the ratios of forces or moments of forces describing a given dynamic problem. Because the dimensions of such quantities are similar, their ratios are dimensionless and as such ones constitute specific criteria/numbers of dynamic similarity of the analyzed issues. This is what this work is about. It consists of three parts.

Part I concerns the bases of dynamic similarity criteria formulated in such a way in relation to the problems of fluid–solid interaction at different fluid–solid relative motions. At the end of this part, authorial method and procedure for determination of dynamic similarity criteria in fluid–solid interaction issues have been presented.

Part II concerns the determination and analysis of dynamic similarity criteria for various practical problems encountered mainly in the hydraulics and fluid dynamics at steady, smooth fluid on-flow in front of a solid. Moreover, the cases of mechanically induced vibrations of a body in a stationary fluid and a fluid moving with constant velocity in front of the body have been presented.

Part III concerns determination and analysis of dynamic similarity criteria for simple cases of vibrations caused by wind encountered in aerodynamics of buildings and structures, including self-exciting vibrations and vibrations caused by turbulent wind.

2 A systemic approach to physical phenomena occurring in mechanics of continuous or discrete material mediums

Material mediums in mechanics usually are divided into solid body (stiff or deformable) and fluids (liquids and gases). Physical phenomena in mechanics have cause and effect character.

If for the processes or phenomena related to some material system (object) there is a cause and effect relationship (or relationships), then the block of data related to the cause is called the input (IN), the block of data related to the material system (object), which is the subject of input influence is called the system, characteristic object or simply object (O), and the third block of data related to effect of that influence is called the output (OU). In mechanics of material mediums the input is often called action, load or force acting on the system; the system (object) may be called the system, structure, construction, etc. and the output is called response or reaction of the system. Every set of these three blocks of data with one input, one object and one output we will call simple system (comp. Fig. 1).

Fig. 1
figure 1

Block diagram of simple system (with one input and one output)

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If several inputs IN act on the system O and there are also several outputs OU, such a system we call complex system with several inputs and several outputs. Likewise, each of the series connection, parallel connection or series–parallel connection of the simple systems create complex series, parallel or series–parallel systems. It is also possible to make complex mixed systems (e.g. series system including subsystems with a large number of inputs and outputs). Issues of mechanics of material mediums can be classified just as input/output complex mixed systems.

Inputs and outputs quantities are sometimes dependent on each other. When system output quantities can influence on system input quantities, then that system is called system with feedback. The aerodynamic feedback which occurs between building vibrations and wind actions on the building is an example of feedback. Building vibrations change character of air flow around the building, and thereby they change distribution of wind pressures on the walls of the building.

Examples of different complex input/output systems are shown schematically in Fig. 2.

Fig. 2
figure 2

Examples of block diagrams of complex input/output systems: a simple system with feedback, b system with several inputs and outputs, c parallel system with several inputs and outputs, d series–parallel system with feedback and several inputs and outputs [13]

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For example, system shown in Fig. 2c can be interpreted as follows: subset O1, is the domain of ground foundations adjacent to building foundations; input IN1 represents vibrations of ground foundations on the part of outer surface of this domain from the vibration source direction, which can be seismic or para-seismic excitations; output OU1 represents accelerations (displacements, velocities) of building foundations, which in turn represent cinematic excitation of the building itself, i.e. object O; input IN2 is for example wind velocity field in front of the building, in the part of outer surface of air domain which is adjacent to the building, so subset O2; output OU2 represents wind pressure field on the walls of the building, what in turn is wind action on the building itself.

Description of the relations occurring within a system with the use of the formal mathematical language and mathematical models of individual study subjects making up the system is called the mathematical model of this system.

An example of the mathematical model of a system adopted fairly commonly in the finite element method (FEM) with reference to linear-elastic systems with viscous damping is the following matrix movement equation:

$${\varvec{M}}\ddot{\varvec{q}} + {\varvec{C}}\dot{\varvec{q}} + {\varvec{Kq}} = \varvec{Q},$$
(1)

where M, C, K matrices of the system masses (inertia), damping and rigidity, respectively; \(\ddot{\varvec{q}}\) \(\dot{{\varvec{q}}}\), \(\ddot{{\varvec{q}}}\), q vectors of generalised node accelerations, node velocities and node displacements of the system, respectively; Q vector of generalised node forces (excitations) of the system resulting from node and spanned actions of the system.

When considering the problems related to the statics of such systems, the equation may be simplified and take the following form:

$${\varvec{Kq}} = {\varvec{Q}}.$$
(2)

The above relations refer to one global set of coordinates XYZ.

At description of some mechanical phenomenon, one can distinguish the following groups of geometrical and physical relationships, which we express as mathematical relationships:

  • Geometrical relationships describing initial geometry (configuration) of a single system or particular subsystems of the whole system (e.g. boundary surface equations of solid body, axle geometry equations of particular bars of bar system);

  • Relationships for quantities related to the restrictions imposed on the system/subsystems, which results from the existence of different type of external/internal constraints of the system/subsystems, which restrain their movement/deformations. That can be kinematic constraints, mechanical constraints or out of mechanical constraints;

  • Relationships resulting from initial conditions with respect to excitations (actions) kinematic, mechanical, out of mechanical acting on the system/subsystems;

  • Relationships arising from geometry and mechanical laws, connecting quantities which describe transition of system/subsystems from initial state (initial configuration) into temporary/final state (temporary/final configuration);

  • Relationships relating from imposing on the system/subsystems different types of restrictions connected with their serviceability (serviceability conditions) and safety (limit conditions).

These groups of relationships are associated with specific groups of geometrical and physical quantities (dimensional or dimensionless variables/parameters) dependent on the scale of the phenomenon, dimensional or dimensionless constants independent on a scale of the phenomenon, which in turn represent different subsets {S}s of the initial set {S} of all the quantities characterizing this mechanical phenomenon.

The essence of model investigations of mechanical phenomena consists on performing respective investigations (tests) of that phenomenon in smaller scale, at fulfilment of model similarity criteria of that phenomenon, and on measurement of dimensionless output quantities of particular subsystems or the system as the whole and the transition of the same values of these quantities to the mechanical phenomenon in the natural scale (or in other scale).

3 Forces occurring in mechanics of fluids and solids or material particles

If two mechanical phenomena are dynamically similar, the corresponding forces must be in the same ratio in the two cases. A force vector \({\varvec{F}}\) of a magnitude F may refer to different physical properties of a fluid, a solid or a particle.

Material particles are understood here to be among others: particles of a crushed solid, powders, microparticles of a solid constituting a suspension, air bubbles in water, water droplets in air, snowflakes in air, fine ice crystals in air or in a snow cloud, etc.

In practical applications of fluid–solid (or fluid–particle) interactions, the following forces that may act on a fluid \(\left(f\right)\), solid \(\left(s\right)\) (or particle \(\left(p\right)\)) element can be distinguished using the respective indices or subscripts:

  • Gravity \(\left(g\right)\) forces: \({F}_{gf}\), \({F}_{gs}\);

  • Pressure \(\left(\Delta p\right)\) forces: \({F}_{\Delta pf}\);

  • Surface pressure \(\left(\Delta p\right)\) forces: \({\mathcal{F}}_{\Delta p}\); \({\mathcal{F}}_{\Delta pf}\), \({\mathcal{F}}_{\Delta ps}\): \({\mathbf{F}}_{\Delta pf}=-{\mathbf{F}}_{\Delta ps}\);

  • Viscosity \(\left(v\right)\) forces: \({F}_{vf}\);

  • Surface viscosity \(\left(v\right)\) forces: \({F}_{v}\); \({F}_{vf}\), \({F}_{vs}\): \({\mathbf{F}}_{vf}=-{\mathbf{F}}_{vs}\);

  • Surface roughness/friction \(\left(r\right)\) forces: \({\mathcal{F}}_{r}\); \({\mathcal{F}}_{rf}\), \({\mathcal{F}}_{rs}\): \({\mathbf{F}}_{rf}=-{\mathbf{F}}_{rs}\);

  • Elasticity \(e\) forces: \({F}_{ef}={F}_{Kf}\); \({F}_{es}\): \({F}_{Es}\), \({F}_{Gs}\);

  • Inertia \(\left(i\right)\) forces: \({F}_{if}\), \({F}_{is}\); \({F}_{ifs}\)

  • Surface tension \(\left(t\right)\) forces: \({F}_{tf}\);

  • Damping \(\left(d\right)\) forces: \({F}_{ds}\);

  • Buoyancy \(\left(b\right)\) forces: \({\mathcal{F}}_{b}\); \({\mathcal{F}}_{bf}\), \({\mathcal{F}}_{bs}\): \({{\varvec{F}}}_{bf}=-{{\varvec{F}}}_{bs}\);

  • Wave \(\left(w\right)\) resistance forces: \({\mathcal{F}}_{w}\); \({\mathcal{F}}_{wf}\), \({\mathcal{F}}_{w}\); \({{\varvec{F}}}_{wf}=-{{\varvec{F}}}_{ws}\)

Rising or sinking forces of a solid (or material particles) submerged in a fluid \({F}_{gb}={F}_{gs}-{\mathcal{F}}_{bs}\).

The forces listed above may be surface forces, and thus related to relevant stresses, or volume forces, and thus related to the densities of relevant forces. Both kinds of forces may operate on small surface or volume elements of a fluid and a solid (particle) of characteristic dimensions \({d}_{f}\) and \({d}_{s}\left({d}_{p}\right)\), respectively, or on large surface or volume elements of a fluid and solid (particle), e.g. on the whole solid, of characteristic dimensions \({D}_{f}\) and \({D}_{s}\left({D}_{p}\right)\), respectively.

Let us now consider selected problems of applied mechanics of fluids and solids or material particles, with special emphasis placed on the problems of fluid–solid or fluid–material particles contact when they are in relative motion. In our further considerations we shall assume that the criteria of geometrical similarity of a given problem have been fulfilled both in the natural scale (prototype identified by subscript P) and the model scale (model identified by subscript M, usually made in a smaller scale).

We shall be interested in local forces, related to small material, surface or volume, elements, and global (resultant) forces, related to large material, surface or volume, elements. In the latter case, we will talk about global (resultant) forces and moments of forces. Since each of the forces or moments of forces is in general a vector quantity, i.e. determined by three components in the Cartesian coordinate system (x, y, z), all or selected components of these forces and moments of forces must be taken into account while considering spatial problems. Some of the problems will refer to one of these components (e.g. air or fluid drag force).

Most of the considered problems are steady problems, i.e. where the velocity of the on-flowing fluid \({V}_{f}\), the characteristic velocity of the solid \({V}_{f}\) (particle \({V}_{p}\)) or their relative velocity are constant in time.

In our considerations we shall additionally assume that the kinematic similarity criteria of the fluid flow on the outer boundary surfaces of the separated fluid volume containing a solid or material particles are fulfilled.

Dynamic similarity criteria, developed and analysed further on, will refer mainly to the following problems:

  • Fluid flows and flows past objects;

  • Fluid–solid contact problems;

  • Problems of a solid floating on a fluid with and without the presence of a surface wave;

  • Lifting and sinking of a solid or material particles in a stationary or moving fluid;

  • Vibrations of a solid in a stationary and moving fluid.

The forces listed above arising in fluids, solids (particles) and on the fluid–solid (fluid–particles) contact surface will be characterised by coefficients or parameters (as a rule of dimensional nature) occurring in dependencies defining these forces. Considering the problem within the boundaries of continuous media mechanics, we shall assume that the forces in question are characterised by the following set of dimensional quantities treating as parameters:

$$\left(\begin{array}{c}g;{\rho }_{f},{\rho }_{s},{\rho }_{p};{V}_{f},{V}_{s},{V}_{p},{V}_{ts},{V}_{tp}{V}_{wf};{d}_{f},{d}_{s},{d}_{p},{D}_{f},{D}_{s},{D}_{p};\\ {\Omega }_{f},{\Omega }_{s},{\Omega }_{p};{K}_{f};{E}_{s},{G}_{s};{\mu }_{f},{\mu }_{s},{\mu }_{r};\Delta {p}_{f}={p}_{f}-{p}_{stf};{\sigma }_{tf},{A}_{wf},{\lambda }_{wf}\end{array}\right),$$
(3)

where

\(g:\) gravitational acceleration; \({\rho }_{f}\), \({\rho }_{s}\), \({\rho }_{p}\): fluid, solid and particles mass density; \({V}_{f}\), \({V}_{s}\), \({V}_{p}\): characteristic velocities of fluid (usually far ahead of the solid), solid and particles; \({V}_{ts}\), \({V}_{tp}\): so-called terminal velocities of solids and particles when they fall freely (or are lifted) in a fluid; \({V}_{wf}\): fluid surface wave velocity; \({d}_{f}\), \({d}_{s}\), \({d}_{p}\), \({D}_{f}\), \({D}_{s}\), \({D}_{p}\): characteristic dimensions of a small or large, surface or volume, element of a fluid, solid and particles; \({\Omega }_{f}\), \({\Omega }_{s}\), \({\Omega }_{p}\): volumes of a large volume elements of a fluid, whole solid and whole particle; \({K}_{f}\): fluid compressibility (bulk elasticity) modulus; \({E}_{s}\), \({G}_{s}\): solid (Young’s) linear and (Kirchhoff’s) shear elasticity modulus; \({\mu }_{f}\): fluid dynamic viscosity coefficient; \({\mu }_{s}\): a solid vibrations (internal friction) damping coefficient; \({\mu }_{r}\): a solid and fluid surface friction coefficient (dimensionless); \({p}_{fs}\): pressure on the fluid element or on the contact surface between a solid and fluid; \({p}_{stf}\): static pressure in the fluid far ahead of the solid; \({\sigma }_{tf}\): fluid surface tension coefficient if the fluid element is at the liquid–gas interface; \({A}_{wf}\), \({\lambda }_{wf}\): fluid surface wave amplitude and length, respectively.

Where a solid (or particles) is treated as a whole (i.e. globally), relevant parameters characterising the global (resultant or net) forces or moments will be specified more precisely further on in our considerations.

In many fluid–solid (fluid–particle) interaction problems, some of the considered forces are either not present or insignificant. In Fig. 3 are depicted two geometrically similar flow systems, i.e. an elastically supported prismatic body of horizontal axis \(y\) normal to the inflowing fluid of velocity \({V}_{f}\). Let it be assumed that they also possess kinematic similarity, and that the global (fluid–particle) interaction forces acting on the body are \({F}_{gs}\), \({\mathcal{F}}_{\Delta ps}\), \({\mathcal{F}}_{rs}\) and \({F}_{es}\). Then dynamic similarity will be achieved if:

Fig. 3
figure 3

Prototype (a) and model (b) of length scale \({k}_{L}=\frac{{D}_{P}}{{D}_{M}}=\frac{{B}_{P}}{{B}_{M}}\) and velocity scale \({k}_{Vf}=\frac{{V}_{fP}}{{V}_{fM}}\), and corresponding forces

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$$\frac{{F}_{gsP}}{{F}_{gsM}}=\frac{{\mathcal{F}}_{\Delta psP}}{{\mathcal{F}}_{\Delta psM}}=\frac{{\mathcal{F}}_{rsP}}{{\mathcal{F}}_{rsM}}=\frac{{F}_{esP}}{{F}_{esM}},$$
(4)

where

subscripts P and M refer to the prototype and the model as before.

These relations can be expressed as

$${\left(\frac{{F}_{es}}{{F}_{gs}}\right)}_{M}={\left(\frac{{F}_{es}}{{F}_{gs}}\right)}_{P},{\left(\frac{{F}_{es}}{{\mathcal{F}}_{\Delta ps}}\right)}_{M}={\left(\frac{{F}_{es}}{{\mathcal{F}}_{\Delta sp}}\right)}_{P},{\left(\frac{{F}_{es}}{{\mathcal{F}}_{rs}}\right)}_{M}={\left(\frac{{F}_{es}}{{\mathcal{F}}_{rs}}\right)}_{P}.$$
(5)

Each of the quantities is dimensionless i.e. it is a similarity number. With four forces acting, there are three independent expressions that must be satisfied; for three forces, there are two independent expressions, and so on. The significance of the dimensionless ratios is discussed in the following paragraphs.

4 Dimensional analysis of individual forces

Let us introduce the following designations and definitions:

  • Any component of force vector or force moment, vector we denote as Q and its dimension as [Q];

  • Any dimensional generalised parameter X characterizing quantity Q in the form of generalized factor or coefficient can be presented in the form of powers of other dimensional parameters \({X}_{i}\) characterizing Q, i.e.

$$X={X}_{1}^{{\beta }_{1}}\dots {X}_{j}^{{\beta }_{j}},$$
(6)

where \({\beta }_{i}\) powers, real numbers, i = 1,…,j;

  • According to the principles applied in dimensional analysis, dimensions of quantities Q and X, i.e. [Q] and [X], may be given by

$$\left[Q\right]={[{A}_{1}]}^{{\alpha }_{1}}\dots {\left[{A}_{k}\right]}^{{\alpha }_{k}},$$
(7)
$$\left[X\right]={[{A}_{1}]}^{{\gamma }_{1}}\dots {\left[{A}_{k}\right]}^{{\gamma }_{k}}={\left[{X}_{1}\right]}^{{\beta }_{1}}\dots {\left[{X}_{j}\right]}^{{\beta }_{j}},$$
(8)
$$\left[{X}_{i}\right]={[{A}_{1}]}^{{\gamma }_{i1}}\dots {\left[{A}_{k}\right]}^{{\gamma }_{ik}},$$
(9)
$$\frac{\left[Q\right]}{[X]}={[{A}_{1}]}^{\left({{\alpha }_{1}-\gamma }_{1}\right)}\dots {\left[{A}_{k}\right]}^{\left({{\alpha }_{k}-\gamma }_{k}\right)}.$$
(10)

Finally,

$$\left[Q\right]={[X][{A}_{1}]}^{{\delta }_{1}}\dots {\left[{A}_{k}\right]}^{{\delta }_{k}}={\left[{X}_{1}\right]}^{{\beta }_{1}}\dots {{\left[{X}_{j}\right]}^{{\beta }_{j}}\left[{A}_{1}\right]}^{{\delta }_{1}}\dots {\left[{A}_{k}\right]}^{{\delta }_{k}}.$$
(11)

Above dimensional relationship can be substituted by the following algebraic relationship, which is equivalent with respect to dimensions and marked by denotation \(\triangleq\):

$$Q\triangleq {X}_{1}^{{\beta }_{1}}\dots {X}_{j}^{{\beta }_{j}}{A}_{1}^{{\delta }_{1}}\dots {A}_{k}^{{\delta }_{k}}.$$
(12)

All powers appeared in power–law relationships are real numbers.

In classical mechanical issues k = 3. Therefore, the relationship (12) takes the form:

$$Q\triangleq {X}_{1}^{{\beta }_{1}}\dots {X}_{j}^{{\beta }_{j}}{A}_{1}^{{\delta }_{1}}{A}_{1}^{{\delta }_{2}}{A}_{k}^{{\delta }_{3}}.$$
(13)

Such cases will be considered further. Moreover, in many cases, \({X=X}_{1}\).

In further considerations, for all the dimensional quantities in set (3) characteristic for individual forces we shell assume a dimensional base of three components: \(\left(\rho ,V,D\right)\) depending on the case under consideration, they will be as follows:

$$\rho = \left\{ {\begin{array}{*{20}l} {\rho_{f} } \\ {\rho_{s} } \\ {\rho_{p} } \\ \end{array} } \right.;V = \left\{ {\begin{array}{*{20}c} {V_{f} } \\ {V_{s} ,V_{p} } \\ {V_{ts} ,V_{tp} } \\ {V_{wf} } \\ \end{array} } \right., D = \left\{ {\begin{array}{*{20}c} {d_{f} ,D_{f} } \\ {d_{s} ,D_{s} } \\ {d_{p} ,D_{p} } \\ \end{array} } \right..$$
(14)

To simplify the notation, further on we shall replace indices s (solid) and p (particle) with one index o (object). Similarly, for components \(x\), \(y\) and \(z\) of global forces and moments, we will also introduce one index j, so

$$o = s, p;\,j = x, y, z.$$

The dimensions of individual local or global forces may thus be expressed by the following dimensional dependencies [17, 18]:

  • Local gravity forces

$$\left[{F}_{g}\right]=\left\{\begin{array}{c}\left[{F}_{gf}\right]={\left[g\right]}^{1}{\left[{\rho }_{f}\right]}^{1}{\left[{d}_{f}\right]}^{3}\\ \left[{F}_{go}\right]={\left[g\right]}^{1}{\left[{\rho }_{o}\right]}^{1}{\left[{d}_{o}\right]}^{3}\end{array}.\right.$$
(15)

  • Global gravity forces

$$\left[{F}_{gz}\right]=\left\{\begin{array}{c}\left[{F}_{gfz}\right]={\left[g\right]}^{1}{\left[{\rho }_{f}\right]}^{1}\left[{\Omega }_{f}\right]={\left[\rho \right]}^{1}{\left[{\rho }_{f}\right]}^{1}{\left[{D}_{f}\right]}^{3}\\ \left[{F}_{goz}\right]={\left[g\right]}^{1}{\left[{\rho }_{o}\right]}^{1}\left[{\Omega }_{o}\right]={\left[\rho \right]}^{1}{\left[{\rho }_{o}\right]}^{1}{\left[{D}_{o}\right]}^{3}\end{array}\right..$$
(16)

  • Local pressure forces

$$\left[{F}_{\Delta pf}\right]={\left[{\Delta p}_{f}\right]}^{1}{\left[{d}_{f}\right]}^{2}.$$
(17)

  • Local surface pressure forces

$$\left[{\mathcal{F}}_{\Delta po}\right]={\left[{\Delta p}_{o}\right]}^{1}{\left[{d}_{o}\right]}^{2}.$$
(18)

  • Global surface pressure forces and moments

  • aerodynamic or hydrodynamic forces

$$\left[{F}_{\Delta poj}\right]={\left[{\Delta p}_{o}^{*}\right]}^{1}{\left[{D}_{o}\right]}^{2}.$$
(19)

  • aerodynamic or hydrodynamic moment

$$\left[{M}_{\Delta poj}\right]={\left[{\Delta p}_{o}^{*}\right]}^{1}{\left[{D}_{o}\right]}^{3},$$
(20)

where

\({\Delta p}_{o}^{*}\) is the characteristic value of \({\Delta p}_{o}\) (e.g. maximum positive pressure).

  • Local and global viscosity forces

$$\left[{F}_{vf}\right]=\left\{\begin{array}{c}{\left[{\mu }_{f}\right]}^{1}{\left[{V}_{f}\right]}^{1}{\left[{d}_{f}\right]}^{1}\\ {\left[{\mu }_{f}\right]}^{1}{\left[{V}_{f}\right]}^{1}{\left[{D}_{f}\right]}^{1}\end{array}.\right.$$
(21)

  • Local surface viscosity forces

$$\left[{\mathcal{F}}_{vo}\right]=\left\{\begin{array}{c}{\left[{\mu }_{f}\right]}^{1}{\left[{V}_{f}\right]}^{1}{\left[{d}_{o}\right]}^{1}\\ {\left[{\mu }_{f}\right]}^{1}{\left[{V}_{o}\right]}^{1}{\left[{d}_{o}\right]}^{1}\end{array}\right..$$
(22)

  • Local surface roughness / friction forces

$$\left[{\mathcal{F}}_{ro}\right]={\left[{\mu }_{ro}\bullet {\Delta p}_{o}\right]}^{1}{\left[{d}_{o}\right]}^{2}.$$
(23)

  • Global surface roughness/friction forces and moments

$$\left[{F}_{roj}\right]={\left[{\mu }_{ro}^{*}\bullet {\Delta p}_{o}^{*}\right]}^{1}{\left[{D}_{o}\right]}^{2}.$$
(24)
$$\left[{M}_{roj}\right]=\left[{\mu }_{ro}^{*}\bullet {\Delta p}_{o}^{*}\right]{\left[{D}_{o}\right]}^{3},$$
(25)

where

\({\mu }_{ro}^{*}\) is the dimensionless characteristic (e.g. mean) roughness/friction coefficient; \({\Delta p}_{o}^{*}\): characteristic value of \(\Delta p\);

  • Local elasticity forces

  • in fluid

$$\left[{F}_{Kf}\right]={\left[{K}_{f}\right]}^{1}{\left[{d}_{f}\right]}^{2}$$
(26)

  • in a solid

$$\left[ {F_{Es} } \right] = \left[ {E_{s} } \right]^{1} \left[ {d_{s} } \right]^{2} ;\left[ {F_{Gs} } \right] = \left[ {G_{s} } \right]^{1} \left[ {d_{s} } \right]^{2} .$$
(27)

  • Global elasticity force and moment

If we consider translational–rotational vibrations of a solid of two degrees of freedom: shear (translational) \(\zeta\) and torsional (rotational) \(\varepsilon\), elastically supported and immersed in a stream of fluid, instead of quantities \({E}_{s}\) and \({G}_{s}\), we could also assume the following quantities characterising the global elasticity force and global elasticity moment of the solid:

  • translational \({k}_{z}\) and torsional \({k}_{\varepsilon }\) rigidity coefficients,

  • normal translational \({f}_{z}\) and torsional \({f}_{\varepsilon }\) vibrations frequencies.

In such case, we will have:

  • for global translational elasticity force,

$$\left[ {F_{kz} } \right] = \left[ {k_{z} } \right]^{1} \left[ {D_{s} } \right]^{1} ;\left[ {F_{fz} } \right] = \left\{ {\begin{array}{*{20}c} {\left[ {f_{z} } \right]^{2} \left[ {\rho_{s} } \right]\left[ {D_{s} } \right]^{4} } \\ {\left[ {f_{z} } \right]^{2} \left[ {\rho_{s} } \right]\left[ {\Omega_{s} } \right]\left[ {D_{s} } \right]} \\ \end{array} } \right.$$
(28)

  • for global torsional elasticity moment,

$$\left[ {M_{k\varepsilon } } \right] = \left[ {k_{\varepsilon } } \right]^{1} ;\left[ {M_{k\varepsilon } } \right] = \left\{ {\begin{array}{*{20}c} {\left[ {f_{\varepsilon } } \right]^{2} \left[ {\rho_{s} } \right]\left[ {D_{s} } \right]^{4} } \\ {\left[ {f_{\varepsilon } } \right]^{2} \left[ {\rho_{s} } \right]\left[ {\Omega_{s} } \right]\left[ {D_{s} } \right]} \\ \end{array} } \right..$$
(29)

  • Cross-sectional elasticity bending moment and torsional moment.

On the other hand, if we consider translational–rotational vibrations of a beam, whose bending-torsional elasticity properties are characterised by bending rigidity \(EJ\) and torsional rigidity \(G{J}_{t}\), where: \(J\), \({J}_{t}\) moments of inertia of a cross-section of a beam when subjected to bending and torsion, we will obtain:

$$\left[ {M_{EJ} } \right] = \left[ E \right]\left[ J \right]\left[ {D_{s} } \right]^{ - 2} ;\,\left[ {M_{GJt} } \right] = \left[ G \right]\left[ {J_{t} } \right]\left[ {D_{s} } \right]^{ - 2} .$$
(30)

  • Local and global inertia forces

  • in fluid

$$\left[{F}_{if}\right]=\left\{\begin{array}{c}{\left[{\rho }_{f}\right]}^{1}{\left[{V}_{f}\right]}^{2}{\left[{d}_{f}\right]}^{2}\\ {\left[{\rho }_{f}\right]}^{1}{\left[{V}_{f}\right]}^{2}{\left[{D}_{f}\right]}^{2}\end{array}\right.$$
(31)

  • in a solid or particle

$$\left[{F}_{io}\right]=\left\{\begin{array}{c}{\left[{\rho }_{o}\right]}^{1}{\left[{V}_{o}\right]}^{2}{\left[{d}_{o}\right]}^{2}\\ {\left[{\rho }_{o}\right]}^{1}{\left[{V}_{o}\right]}^{2}{\left[{D}_{o}\right]}^{2}\end{array}\right.$$
(32)

  • coming from the so-called added fluid mass moving jointly (co-vibrating) with the solid or particle.

When a solid (or particle) immersed in a fluid experiences acceleration, e.g. in a vibratory motion some fluid mass connected with the solid performs motions that are similar to the motions of the solid. This additional fluid mass connected with the solid is called added mass or virtual mass or hydrodynamic mass. Additional forces of fluid inertia \({F}_{ifo}\) arise at this situation, called added mass forces, which affect the solid. And thus we will have:

$$\left[{F}_{ifo}\right]=\left\{\begin{array}{c}{\left[{\rho }_{f}\right]}^{1}\left[{V}_{o}\right]{\left[{d}_{o}\right]}^{2}\\ {\left[{\rho }_{f}\right]}^{1}\left[{V}_{o}\right]{\left[{D}_{o}\right]}^{2}\end{array}.\right.$$
(33)

  • Global inertia forces and moments for a vibrating solid and added fluid mass

If we consider again, like in the previous case, translational–rotational vibrations of a solid body of two degrees of freedom: translational \(\zeta\) and rotational \(\varepsilon\), immersed in a fluid, instead of quantities \({\rho }_{s}\) and \({\rho }_{f}\), we could assume the following quantities characterising the global translational inertia forces and inertia moments as well as the inertia force and moment coming from the added mass:

  • translational inertia force of a solid body \({F}_{isz}\)

$$\left[{F}_{isz}\right]=\left\{\begin{array}{c}{\left[m\right]}^{1}{\left[{V}_{s}\right]}^{2}{\left[{D}_{s}\right]}^{-1}\\ {\left[{\rho }_{s}\right]}^{1}{\left[{\Omega }_{s}\right]}^{1}{\left[{V}_{s}\right]}^{2}{\left[{D}_{s}\right]}^{-1}\end{array}\right.$$
(34)

  • rotational inertia force of a solid body \({M}_{is\varepsilon }\)

$$\left[{M}_{is\varepsilon }\right]=\left\{\begin{array}{c}{\left[J\right]}^{1}{\left[{V}_{s}\right]}^{2}{\left[{D}_{s}\right]}^{-2}\\ {\left[{\rho }_{s}\right]}^{1}{\left[{\Omega }_{s}\right]}^{1}{\left[{V}_{s}\right]}^{2}\end{array}\right.$$
(35)

  • added mass force associated with translation of a solid body \({F}_{ifsz}\)

$$\left[{F}_{ifsz}\right]=\left\{\begin{array}{c}{\left[{m}_{f}\right]}^{1}{\left[{V}_{s}\right]}^{2}{\left[{D}_{s}\right]}^{-1}\\ {\left[{\rho }_{f}\right]}^{1}{\left[{\Omega }_{s}\right]}^{1}{\left[{V}_{s}\right]}^{2}{\left[{D}_{s}\right]}^{-1}\end{array}\right.,$$
(36)

  • added mass moment associated with rotation of a solid body \({M}_{ifs\varepsilon }\)

$$\left[{M}_{ifs\varepsilon }\right]=\left\{\begin{array}{c}{\left[{J}_{f}\right]}^{1}{\left[{V}_{s}\right]}^{2}{\left[{D}_{s}\right]}^{-2}\\ {\left[{\rho }_{f}\right]}^{1}\left[{\Omega }_{s}\right]{\left[{V}_{s}\right]}^{2}\end{array},\right.$$
(37)

  • net inertia force on a solid body \({F}_{iz\Sigma }={F}_{isz}+{F}_{ifsz}.\)

$$\left[{F}_{iz\Sigma }\right]={\left[{\rho }_{s}+{\rho }_{f}\right]}^{1}{\left[{\Omega }_{s}\right]}^{1}{\left[{V}_{s}\right]}^{2}{\left[{D}_{s}\right]}^{-1},$$
(38)

  • net inertia moment on a solid body \({M}_{i\varepsilon \Sigma }={M}_{is\varepsilon }+{M}_{ifs\varepsilon },\)

$$\left[{M}_{i\varepsilon \Sigma }\right]={\left[{\rho }_{s}+{\rho }_{f}\right]}^{1}{\left[{\Omega }_{s}\right]}^{1}{\left[{V}_{s}\right]}^{2}.$$
(39)

The following additional designations have been used in the above relations: \(m\) solid body mass; \(J\) solid body rotational inertia moment; \({m}_{f}\) added fluid mass; \({J}_{f}\) rotational inertia moment of the added fluid mass; \({\Omega }_{s}\) solid body volume.

  • Local surface tension forces

$$\left[{F}_{tf}\right]=\left[{\sigma }_{tf}\right]\left[{d}_{f}\right].$$
(40)

  • Local damping forces

$$\left[{F}_{ds}\right]={\left[{\mu }_{s}\right]}^{1}{\left[{V}_{s}\right]}^{1}{\left[{d}_{s}\right]}^{2}.$$
(41)

  • Global damping force and damping force moment

Let us consider again, as above, translational–rotational vibrations of a solid immersed in a fluid. Assuming that the vibrations damping force and moment are described by linear models of the viscous type (i.e. proportional to the translational and rotational vibrations velocities \(\dot{z}\) and \(\dot{\varepsilon }\), respectively), we could introduce the following translational and rotational vibrations damping coefficients – \({\mu }_{sz}\) and \({\mu }_{s\varepsilon }\): \({\mu }_{szs}\), \({\mu }_{szf}\) translational damping force coefficient coming from the solid and the fluid, respectively, (the so-called hydrodynamic or aerodynamic damping forces); \({\mu }_{s\varepsilon s}\), \({\mu }_{s\varepsilon f}\) as above, but with reference to the rotational damping moment; \({\mu }_{sz\Sigma }={\mu }_{szs}+{\mu }_{szf}\) resultant translational damping coefficient; \({\mu }_{s\varepsilon \Sigma }={\mu }_{s\varepsilon s}+{\mu }_{s\varepsilon f}\) resultant rotational damping coefficient. Then the following will be true:

  • for global translational vibrations damping force \({F}_{dz\Sigma }\)

$$\left[{F}_{dz\Sigma }\right]={\left[\left({\mu }_{szs}+{\mu }_{szf}\right)\right]}^{1}{\left[{V}_{s}\right]}^{1}$$
(42)

  • for global rotational vibrations damping moment \({M}_{d\varepsilon \Sigma }\)

$$\left[{M}_{d\varepsilon \Sigma }\right]={\left[\left({\mu }_{s\varepsilon s}+{\mu }_{s\varepsilon f}\right)\right]}^{1}\left[{V}_{s}\right]{\left[D\right]}^{-1}.$$
(43)

  • Global buoyancy forces for a solid and a particle

$$\left[{F}_{b}\right]={\left[{\rho }_{f}\right]}^{1}{\left[g\right]}^{1}{\left[{D}_{o}\right]}^{3}.$$
(44)

  • Global wave resistance force for a solid

$$\left[{F}_{w}\right]={\left[{A}_{w}\right]}^{1}{\left[{\lambda }_{w}\right]}^{-1}{\left[{\rho }_{f}\right]}^{1}{\left[{V}_{w}\right]}^{2}{\left[{D}_{s}\right]}^{2}.$$
(45)

  • Global uplifting or falling forces of a solid or material particles immersed in a fluid

$$\left[{F}_{gb}\right]=\left\{\begin{array}{c}{\left[\left({\rho }_{s}-{\rho }_{f}\right)\right]}^{1}{\left[g\right]}^{1}{\left[{D}_{o}\right]}^{3}\\ {\left[\left({\rho }_{s}-{\rho }_{f}\right)\right]}^{1}{\left[{V}_{t}\right]}^{2}{\left[{D}_{o}\right]}^{2}\end{array}\right..$$
(46)

5 Model similarity scales of physical phenomena: scale ratios

From dimensional analysis, one can also derived certain general relationships, which can be complied in the case of scales of physical quantities characterizing some physical phenomenon. Therefore, if set of these physical quantities is denoted as (Qp, Qq, Qr; Q1, Q2,…,Qn), where (Qp, Qq, Qr) form dimensional base of phenomenon, i.e. dimensionally independent quantities, which contain dimensions of basic base M for mass, L for length and T for time (e.g. in mechanical problems), the following relationships can be written:

$$Q_{1} = Q_{p}^{{\alpha_{1} }} Q_{q}^{{\beta_{1} }} Q_{r}^{{\gamma_{1} }} ; \ldots ; Q_{n} = Q_{p}^{{\alpha_{n} }} Q_{q}^{{\beta_{n} }} Q_{r}^{{\gamma_{n} }} .$$
(47)
$${\left(\frac{{Q}_{1}}{{Q}_{p}^{{\alpha }_{1}}{Q}_{q}^{{\beta }_{1}}{Q}_{r}^{{\gamma }_{1}}}\right)}_{M}={\Pi }_{1M}={\Pi }_{1P}={\left(\frac{{Q}_{1}}{{Q}_{p}^{{\alpha }_{n}}{Q}_{q}^{{\beta }_{n}}{Q}_{r}^{{\gamma }_{n}}}\right)}_{P},$$
(48)
$${\left(\frac{{Q}_{n}}{{Q}_{p}^{{\alpha }_{n}}{Q}_{q}^{{\beta }_{n}}{Q}_{r}^{{\gamma }_{n}}}\right)}_{M}={\Pi }_{nM}={\Pi }_{nP}={\left(\frac{{Q}_{n}}{{Q}_{p}^{{\alpha }_{n}}{Q}_{q}^{{\beta }_{n}}{Q}_{r}^{{\gamma }_{n}}}\right)}_{P},$$
(49)

or

$$\frac{{Q}_{1M}}{{Q}_{1P}}={k}_{{Q}_{1}}=\frac{{\left({Q}_{p}^{{\alpha }_{1}}{Q}_{q}^{{\beta }_{1}}{Q}_{r}^{{\gamma }_{1}}\right)}_{M}}{{\left({Q}_{p}^{{\alpha }_{1}}{Q}_{q}^{{\beta }_{1}}{Q}_{r}^{{\gamma }_{1}}\right)}_{P}}={k}_{{Q}_{p}}^{{\alpha }_{1}}{k}_{{Q}_{q}}^{{\beta }_{1}}{k}_{{Q}_{r}}^{{\gamma }_{1}} \frac{{Q}_{nM}}{{Q}_{nP}}={k}_{{Q}_{n}}=\frac{{\left({Q}_{p}^{{\alpha }_{n}}{Q}_{q}^{{\beta }_{n}}{Q}_{r}^{{\gamma }_{n}}\right)}_{M}}{{\left({Q}_{p}^{{\alpha }_{n}}{Q}_{q}^{{\beta }_{n}}{Q}_{r}^{{\gamma }_{n}}\right)}_{P}}= {=k}_{{Q}_{p}}^{{\alpha }_{n}}{k}_{{Q}_{q}}^{{\beta }_{n}}{k}_{{Q}_{r}}^{{\gamma }_{n}},$$
(50)

where

\({\Pi }_{1},\dots ,{\Pi }_{n}\) similarity numbers; \(M\) subscript denoting model scale; \(P\) subscript denoting prototype (nature) scale; \({k}_{Qp}, {k}_{Qq},{k}_{Qr}\) scales of dimensional base quantities; \({k}_{Q1},\dots {k}_{Qn}\) scales of the other dimensional dependent quantities.

Therefore, it is assumed that dimensional base of some issue represent following quantities: velocity \(v\), length L and density ρ (or also reference quantities \(v\) o, Lo, ρo), then, transferring the results of measurements obtained from model to the object in natural scale, the following relations between scales of dimensional base \({k}_{v}\), \(k\) L, \(k\) ρ and scales the other physical quantities characterizing that phenomenon should be used:

  • the scale of actions \(k\)P:

$$[P] = [\upsilon ][L]^{2} [\rho ]^{1} ;{ }k_{P} = k_{v}^{2} k_{L}^{2} k_{\rho } ,$$
(51)

  • the scale of pressures (stresses) \(k_{P}\):

$$[p] = [P]^{1} [L]^{ - 2} ;k_{p} = k_{v}^{2} k_{\rho }$$
(52)

  • the scale of time \(k\)t:

$$\left[ t \right] = \left[ \upsilon \right]^{ - 1} \left[ L \right]^{{1}} ;k_{v}^{ - 1} k_{L} ,$$
(53)

  • the scale of frequency \(k\)f:

$$\left[ f \right] = [\upsilon ]^{1} \left[ L \right]^{ - 1} ;k_{f} = 1/k_{t} = k _{L}^{ - 1} ,$$
(54)

  • the scale of mass density per unit length of element \(k\) m:

$$\left[ m \right] = \left[ L \right]^{{2}} \left[ \rho \right]^{{1}} ; k_{m} = k_{L}^{2} k_{\rho } ;$$
(55)

  • the scale of moment of mass inertia density per unit length of element \(k\) mb:

$$\left[ {m_{b} } \right] = \left[ L \right]^{{4}} \left[ \rho \right]^{{1}} ;k_{mb} = k_{L}^{{4}} k_{\rho } ;$$
(56)

  • the scale of longitudinal rigidity \(k\) EA, flexural rigidity \(k\) EI and torsional rigidity \(k\) GIs:

$$\left[ {EA} \right] = \left[ \upsilon \right]^{{2}} \left[ L \right]^{{2}} \left[ \rho \right];k_{EA} = k_{v}^{2} k_{L}^{{2}} k_{\rho } ;$$
(57)
$$[EI] = \left[ \upsilon \right]^{2} [L]^{{4}} [\rho ];k_{EI} =^{ } k_{v}^{2} k_{L}^{4} k_{\rho } ;$$
(58)
$$[GI_{s} ] = \left[ \upsilon \right]^{2} [L]^{{4}} [\rho ];k_{GIs} = k_{v}^{2} k_{L}^{4} k_{\rho } = k_{EI} .$$
(59)

In Part II and Part III, selected individual cases of forces and moments of forces ratios will be presented in detail and analysed. It follows from these considerations that the Reynolds number, the Froude number and the Mach number are the similarity numbers most commonly encountered in fluid mechanics, wind engineering and aerodynamics of building and structures. Scale ratios for different quantities resulting from fulfilment of these similarity numbers are presented in Table 1. These enable quick calculation of the scale ratio (prototype divided by model) or of any desired quantity as long as the given dimensionless number is the same in both prototype and model. Obviously, the computed ratio gives a realistic result only if the flow is predominantly governed by the specific dimensionless number in question.

Table 1 Flow characteristics and similitude scale ratios (ratio of the prototype quantity to the model quantity) [5]
Full size table

6 A. Flaga’s method and procedure for determining dynamic similarity criteria in different issues of fluid–solid interaction, i.e. at different fluid–solid relative motions

This method and procedure is described as follows:

  1. 1.

    In formulas determining components of force vectors or force moment vectors occur different quantities, which could be numbers, parameters or variables. These quantities could be dimensional or dimensionless.

  2. 2.

    Variables appeared in expressions defining vector components as above can be independent variables as variables of space position (x,y,z) or/and of time t or/and dependent variables as functions or generalized functions of independent variables (x,y,z,t), or/and dependent variables depending on parameter sets of input (IN), object (O) and output (OU).

  3. 3.

    From all set of dimensional parameters (\({X}_{1},\dots , {X}_{j})\), characteristic for particular component of force or force moment vector, we select subset of dimensional base parameters (\({A}_{1},\dots , {A}_{k})\). In classical engineering issues it is subset of three parameters, i.e. k = 3. Such cases will be considered further.

  4. 4.

    For dimensional independent variables x,y,z,t, we define dimensionless independent variables \(\check{x}, \check{y},\check{z},\check{t}\) making use of the dimensional base (\({A}_{1},{A}_{2}, {A}_{3})\). For example, if we assume dimensional base as \((\rho ,V,D)\), it would be:

    $$\check{x} = \frac{x}{D},\check{y} = \frac{y}{D},\check{z} = \frac{z}{D},\check{t} = \frac{V}{D}t$$
    (60)
  5. 5.

    Similarity, for all dimensional parameters of sets (IN), (O), (OU), one can determine the following dimensionless parameters sets \((\check{IN})\), \(\left(\check{O}\right), (\check{OU})\).

  6. 6.

    Basing on rules of dimensional analysis, for quantities Q, X, Xi the respective dimensional relationship can be formulated as in chapter 4.

  7. 7.

    In much more cases of fluid–solid interaction issues, any force of a force moment Q could be presented in a form of generalised monomial (or polynomial) as:

    $$Q = C\varphi_{1} (\left( {IN} \right),\left( O \right), \left( {OU} \right))\varphi_{2} \left( {x,y,z} \right)\varphi_{3} \left( t \right)X$$
    (61)

where \(C\) numeric constant (number); \({\varphi }_{1}\left(\dots \right), {\varphi }_{2}\left(\dots \right),{\varphi }_{3}\left(\dots \right)\)- functional relationships, where in general dependences \({\varphi }_{2}\left(x,y,z\right)\) and \({\varphi }_{3}\left(t\right)\) could be common functional functions or generalized dependences of derivative, integral or other character (e.g. as a derivative of system displacements, integral convolutions of output quantities); \(X={X}_{1}^{{\beta }_{1}}\dots {X}_{j}^{{\beta }_{j}}\) (comp. Eq. 6) generalized factor or coefficient characterizing quantity Q. Similar relationship can be formulated for generalized polynomial form as:

$$Q={\sum_{r}{(C{\varphi }_{1}\left(\left(IN\right), \left(O\right), \left(OU\right)\right){\varphi }_{2}\left(x,y,z\right){\varphi }_{3}\left(t\right)X)}_{r}}.$$
(62)

Further consideration will concern generalized monomial form (61). Similar consideration one can perform also for generalized polynomial relationship (62).

  1. 8.

    Making use of dimensional base (\({A}_{1},{A}_{2}, {A}_{3})\), generalized relationship could be brought into another form of this relationship, namely

    $$Q = \check{C}\check{\varphi} _{1} \left( {(\check{IN}),(\check{O}),(\check{OU})} \right),\check{\varphi} _{2} \left( {\check{x},\check{y},\check{z}} \right)\check{\varphi} _{3} (\check{t}) \cdot X \cdot \frac{{\left[ Q \right]}}{{\left[ X \right]}}.$$
    (63)
  2. 9.

    Above dimensional relationship could be substituted by the following generalized relationship, which is equivalent with respect to dimensions, marked by denotation \(\triangleq\) (comp. (10) and (12)):

    $$Q \triangleq \check{C}\check{\varphi} _{1} \left( {(\check{IN}),(\check{O}),(\check{OU})} \right)\varphi _{2} (\check{x},\check{y},\check{z})\mathop \varphi \limits^{} _{3} (t) \cdot X_{1}^{{\beta _{1} }} \ldots X_{j}^{{\beta _{j} }} A_{1}^{{\delta _{1} }} A_{2}^{{\delta _{2} }} A_{3}^{{\delta _{3} }} . .$$
    (64)
  3. 10.

    We could proceed similarity with respect to a reference quantity \({Q}_{ref}\). In such case it will be

    $$\begin{gathered} Q_{{ref}} \triangleq \check{C}_{{ref}} \check{\varphi} _{{1ref}} \left( {\left( {{\check{IN}}_{{ref}} } \right),\left( {\check{O}_{{ref}} } \right),\left( {\check{OU}_{{ref}} } \right)} \right)\check{\varphi} _{{2ref}} (\check{x},\check{y},\check{z})\varphi _{{3ref}} (\check{t}) \cdot \hfill \\ X_{{1ref}}^{{\beta _{{1ref}} }} \ldots X_{{jref}}^{{\beta _{{jref}} }} A_{1}^{{\delta _{{1ref}} }} A_{2}^{{\delta _{{2ref}} }} A_{3}^{{\delta _{{3ref}} }} . \hfill \\ \end{gathered}$$
    (65)
  4. 11.

    In the final step, ratio of quantities Q and \({Q}_{ref}\) determined by relationship (64) and (65) should be created. Such ratio is dimensionless. All dimensionless quantities appearing in such generalized quotient constitute similarity criteria/numbers.

  5. 12.

    In such method and procedure, among others, the following similarity criterion/number can be determined:

    $$\begin{gathered} {\Pi }_{a} = \frac{{X_{1}^{{\beta_{1} }} \ldots X_{1}^{{\beta_{j} }} A_{1}^{{\delta_{1} }} A_{2}^{{\delta_{2} }} A_{3}^{{\delta_{3} }} }}{{X_{1ref}^{{\beta_{1ref} }} \ldots X_{1ref}^{{\beta_{jref} }} A_{1}^{{\delta_{1ref} }} A_{2}^{{\delta_{2ref} }} A_{3}^{{\delta_{3ref} }} }} = \hfill \\ = X_{1}^{{(\beta_{1} - \beta_{1ref} )}} \ldots X_{j}^{{(\beta_{j} - \beta_{jref} )}} A_{1}^{{(\delta_{1} - \delta_{1ref} )}} A_{2}^{{(\delta_{2} - \delta_{2ref} )}} A_{3}^{{(\delta_{3} - \delta_{3ref} )}} . \hfill \\ \end{gathered}$$
    (66)

This authorial method and procedure for determination of dynamic similarity criteria in different fluid–body/particle interaction issues have been presented in the Part II. Basing on this method and procedure, both well-known dynamic similarity criteria, as well as new original dynamic similarity criteria have been obtained and analysed.

7 Dynamic similarity criteria derived from ratios of forces/moments forces occurring in single-degree-of-freedom-system motion equation

For the sake of clarity and additional simplification of the considerations carried out in the second and third part of the paper, it has been assumed that fluid on-flow is a uniform (homogenous) on-flow with respect both to the average fluid velocity and turbulence intensity of fluid velocity fluctuations and that undeformable cylinder of any compact cross section (not necessary circular), supported elastically (including damping) on the ends in the directions x and y is a physical model of the system (i.e. so-called sectional model of a structure—comp, Fig. 4).

$$[{W_x}(t) = \int\nolimits_{ - L/2}^{L/2} {{W_x}(t)dz}]$$
Fig. 4
figure 4

Sectional model of a slender cylindrical structure under fluid onflow excitation

Full size image

For each of directions x and y this system can be treated as one-degree-of-freedom system. Let \(\eta =x,y\) denotes one of these two degrees of freedom.

In general the following relationship for hydro/aero-dynamic action \({w}_{\eta }\) per unit length of the cylinder (i.e. local action) caused by oncoming fluid flow and fluid-body feedbacks, could be written:

$${w}_{\eta }={w}_{\eta }\left(\eta ,\dot{\eta },\ddot{\eta };z,t\right)=qD{\check{w}}_{\eta }\left(\eta ,\dot{\eta },\ddot{\eta };z,t\right);q=\frac{1}{2}\rho {V}^{2},$$
(67)

where \(q\) flow velocity pressure (or flow dynamic pressure); \(\rho\) fluid density;\(V\) velocity of the fluid onflow; \(\eta =\eta \left(z,t\right),\) \(\dot{\eta }=\dot{\eta }\left(z,t\right),\) \(\ddot{\eta }=\ddot{\eta }\left(z,t\right)\)—displacement, velocity, acceleration of the cylinder vibrations, respectively; \({\check{w}}_{\eta }={w}_{\eta }\left(\eta ,\dot{\eta },\ddot{\eta };z,t\right)\) dimensionless local hydro/aero-dynamic action on the cylinder. It should be stressed, however, that for shortening the notation in functional relations, given in this equation in parentheses, only these variables are written in an explicit way which are treated as independent (z,t) or dependent \(\left(\eta ,\dot{\eta },\ddot{\eta }\right)\) variables. The respective sets of parameters characterizing input (IN), object (body) (O) and output (OU) do not appeared in relationship (67). So, to be precise, this functional dependence takes more complicated form, namely

$${w}_{\eta }={w}_{\eta }\left(\left(IN\right),\left(O\right),\left(OU\right);\eta ,\dot{\eta },\ddot{\eta };z,t)\right),$$
(68)
$${\check{w}}_{\eta }={\check{w}}_{\eta }\left(\left(IN\right),\left(O\right),\left(OU\right);\eta ,\dot{\eta },\ddot{\eta };z,t)\right),$$
(69)
$$\eta =\eta \left(\left(IN\right),\left(O\right),\left(OU\right);t\right),$$
(70)
$$\dot{\eta } = \frac{d\eta }{{dt}};\ddot{\eta } = \frac{{d^{2} \eta }}{dt}.$$
(71)

Using shortening notation in functional relationships, the differential equation of motion (vibrations) of the cylinder can be written in the following general form:

$$M\left(\ddot{\eta }+2\gamma {\omega }_{n}\dot{\eta }+{\omega }_{n}^{2}\eta \right)={W}_{\eta }\left(\eta ,\dot{\eta },\ddot{\eta };t\right)=qDL{\check{W}}_{\eta }\left(\eta ,\dot{\eta },\ddot{\eta };t\right),$$
(72)

where M total (generalized, modal) mass of the cylinder (M = mL; where: m mass per cylinder unit length, L – cylinder length); \(\gamma\) coefficient of critical damping (critical damping ratio) for the cylinder (\(\gamma \cong \frac{\Delta }{2\pi }\) where \(\Delta\) logarithmic decrement of vibrations damping); \({\omega }_{n}\) natural (free vibrations) angular frequency of the cylinder (\({\omega }_{n}=2\pi {f}_{n}\) where \({f}_{n}\)—natural frequency or eigenfrequency); \({W}_{\eta }, {\check{W}}_{\eta }\) respectively dimensional and dimensionless global (total, generalized, modal) hydro/aero-dynamic action given by:

$${W}_{\eta }\left(\eta ,\dot{\eta },\ddot{\eta };t\right)={\int }_{-\frac{L}{2}}^\frac{L}{2}{w}_{\eta }\left(\eta ,\dot{\eta },\ddot{\eta };z,t\right)dz=qD{\int }_{-\frac{L}{2}}^\frac{L}{2}{\check{w}}_{\eta }\left(\eta ,\dot{\eta },\ddot{\eta };z,t\right)dz=qDH{\check{W}}_{\eta }\left(\eta ,\dot{\eta },\ddot{\eta },t\right),$$
(73)
$${\check{W}}_{\eta }\left(\eta ,\dot{\eta },\ddot{\eta };t\right)=\frac{1}{L}{\int }_{-\frac{L}{2}}^\frac{L}{2}{\check{w}}_{\eta }\left(\eta ,\dot{\eta },\ddot{\eta };z,t\right)dz.$$
(74)

Assuming dimensional base as \(\left(\rho ,V,D\right)\) the following dimensionless sets of parameters and dimensionless variables could be defined:

  • dimensionless sets of parameters: \((\check{IN})\), \(\left(\check{O}\right), (\check{OU})\)

  • independent variables \(\check{z }\) and \(\check{t }\) defined as:

$$\check{z} = \frac{z}{D}; \check{t} = \frac{{Vt}}{D};dz = Dd\,\check{z};dt = \frac{D}{V}d\check{t},$$
(75)

  • dependent variables \(\check{\eta }\left(\check{t}\right)\); \(\frac{d\check{\eta }(\check{t})}{d\check{t}}\); \(\frac{{d}^{2}\check{\eta }(\check{t})}{{d}^{2}\check{t}}\) given by relationship:

$$\frac{{\eta \left( t \right)}}{D} = \frac{{\eta \left( {\frac{D}{V}\check{t}} \right)}}{D} = \frac{{\eta ^{*} \left( \check{t} \right)}}{D} = \check{\eta} \left( \check{t} \right);\eta \left( t \right) = D\check{\eta} \left( \check{t} \right),$$
(76)
$$d\eta \left(t\right)=d{\eta }^{*}\left(\check{t}\right)=Dd\check{\eta }\left(\check{t}\right),$$
(77)
$${d}^{2}\eta \left(t\right)={d}^{2}{\eta }^{*}\left(\check{t}\right)=D{d}^{2}\check{\eta }\left(\check{t}\right),$$
(78)
$$d{t}^{2}=d{\left(\frac{D}{V}\check{t}\right)}^{2}={\left(\frac{D}{V}\right)}^{2}d{\check{t}}^{2},$$
(79)
$$\dot{\eta }\left(t\right)=\frac{d\eta \left(\frac{D}{V}\check{t}\right)}{dt}=\frac{d{\eta }^{*}\left(\check{t}\right)}{\frac{D}{V}d\check{t}}=\frac{Dd\check{\eta }\left(\check{t}\right)}{\frac{D}{V}d\check{t}}=\frac{V}{D}\frac{d\check{\eta }\left(\check{t}\right)}{d\check{t}},$$
(80)
$$\ddot{\eta }\left(t\right)=\frac{{d}^{2}\eta \left(\frac{D}{V}\check{t}\right)}{{dt}^{2}}=\frac{{d}^{2}{\eta }^{*}\left(\check{t}\right)}{{\left(\frac{D}{V}\right)}^{2}{d\check{t}}^{2}}=\frac{{V}^{2}}{D}\frac{{d}^{2}\check{\eta }\left(\check{t}\right)}{{d\check{t}}^{2}}.$$
(81)

Assuming in the considered case above relationship, motion Eq. (72) can be now written in a dimensionless form as

$$M\left(\frac{{V}^{2}}{D}\frac{{d}^{2}\check{\eta }\left(\check{t}\right)}{{d\check{t}}^{2}}+4\pi \gamma {f}_{n}V\frac{\check{d}\eta \left(\check{t}\right)}{d\check{t}}+4{\pi }^{2}{f}_{n}^{2}D\check{\eta }\left(\check{t}\right)\right)=$$
$$=qDL{\check{W}}_{\eta }\left(D\check{\eta }\left(\check{t}\right),V\frac{d\check{\eta }\left(\check{t}\right)}{{d\check{t}}},\frac{{V}^{2}}{D}\frac{{d}^{2}\check{\eta }\left(\check{t}\right)}{d{\check{t}}^{2}};\frac{D}{V}\check{t}\right)= qDL{\check{W}}_{\eta }^{*}\left(\check{\eta }\left(\check{t}\right),\frac{d\check{\eta }\left(\check{t}\right)}{{d\check{t}}},\frac{{d}^{2}\check{\eta }\left(\check{t}\right)}{d{\check{t}}^{2}};\check{t}\right)$$
(82)

or

$$\frac{{d}^{2}\check{\eta }\left(\check{t}\right)}{{d\check{t}}^{2}}+4\pi \gamma \frac{{f}_{n}D}{V}\frac{d\check{\eta }\left(\check{t}\right)}{d\check{t}}+{4\pi }^{2}{\left(\frac{{f}_{n}D}{V}\right)}^{2}\check{\eta }\left(\check{t}\right)=\frac{D}{M{V}^{2}}qDL{\check{W}}_{\eta }\left(D\check{\eta }\left(\check{t}\right),V\frac{d\check{\eta }\left(\check{t}\right)}{{d\check{t}}},\frac{{V}^{2}}{D}\frac{{d}^{2}\check{\eta }\left(\check{t}\right)}{d{\check{t}}^{2}};\frac{D}{V}\check{t}\right)==\frac{\rho {D}^{2}}{2m}{\check{W}}_{\eta }^{*}\left(\check{\eta }\left(\check{t}\right),\frac{d\check{\eta }\left(\check{t}\right)}{{d\check{t}}},\frac{{d}^{2}\check{\eta }\left(\check{t}\right)}{d{\check{t}}^{2}};\check{t}\right).$$
(83)

In this dimensionless form of motion equation two new dimensionless factors (i.e. dynamic similarity criteria) have appeared, namely

$$Sr=\frac{{f}_{n}D}{V}\mathrm{ called kinematic Strouhal number}$$
(84)
$$Vr=\frac{V}{{f}_{n}D}\mathrm{called reduced velocity}$$
(85)
$$M\rho =\frac{\rho {D}^{2}}{2m}\mathrm{called dimensionless parameter of mass}.$$
(86)

Dimensionless monomials occurring in dimensionless form of motion Eq. (83), constituting dynamic similarity criteria, could be interpreted as the measures of the ratios of the respective hydro/aerodynamic forces, i.e.

  • \(4\pi \gamma Sr\): dimensionless ratio of the parameters resulting from damping force and inertial force,

  • \(4{\pi }^{2}{Sr}^{2}:\)dimensionless ratio of the parameters resulting from elastic force and inertial force.

  • \(M\rho :\)dimensionless ratio of the parameters resulting from hydro-/aero-dynamic force and inertial force.

Moreover, instead of combined parameter of hydro/aero-dynamic action and inertial force \(\frac{{\rho D}^{2}}{2m}\), the combined parameter of hydro/aero-dynamic action and damping force can be defined as

$$\frac{\frac{\rho {D}^{2}}{2m}}{4\pi \gamma Sr}\cong \frac{\rho {D}^{2}}{4m\Delta Sr}=\frac{1}{2}\frac{M\rho }{\Delta }\frac{1}{Sr}=\frac{1}{2}\frac{1}{Sc}\frac{1}{Sr}=\frac{1}{2}\frac{{V}_{r}}{Sc},$$
(87)

where

$$Sc=\frac{\Delta }{{M}_{\rho }}=\frac{2m\Delta }{\rho {D}^{2}}$$
(88)

is the new dimensionless parameter called the Scruton number Sc or the combined parameter of mass and damping.

All dimensionless quantities (i.e. constants, numbers, parameters and independent or dependent variables), constitute dynamic similarity criteria of the considered issue. In model investigations, at least most important from them should be fulfilled.

Dynamic similarity criteria derived as above will be the base of considerations presented in the Part III devoted to determination and analysis of dynamic similarity criteria for simple cases of vibrations caused by wind encountered in aerodynamics of buildings and structures.