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Adv Syst Sci Appl 2023; 01; 99-114 
Published online at https://ijassa.ipu.ru. 

 

Numerical Investigation of Fluid-Structure, Thermal 
Coupling for a Heated Slab 

Oudrane A.1*, Aour B.2, Hamouda M.1, Chesneau X.3, Zeghmati B.3, Balti J.4 
1) Laboratory of Sustainable Development and Informatics (LDDI), Faculty of Science 
and Technology, Ahmed Draya University of Adrar, Algeria 
2) Laboratory of Applied Biomechanics And Biomaterials (LABAB), BP 1523 El 
Mnaouer, National Polytechnic School of Oran-Maurice Audin (ENPO-MA), 31000, 
Oran, Algeria 
3) Laboratory of Mathematics and PhySics Groups of Energy Mechanics (LAMPS), 
university of Perpignan Via Domitia, 52, Avenue Paul Alduy, 66860 Perpignan cedex 
France 
4) Department of Physics, Faculty of Science, University of Carthage, 7021 Jarzouna, 
Tunisia 

Abstract: This work focuses on a numerical analysis of fluid-structure thermal coupling in a heating slab. 
The latter consists of a rectangular cross-section duct located in the concrete slab of a discredited habitat. We 
modeled the thermal transfers of fluid flow in the pipe. In fact, the Navier-Stokes equations that govern this 
flow have been solved numerically. These equations were by an implicit method of finite differences. The 
systems of algebraic equations thus obtained were solved by the algorithms of Gauss and Thomas. The equation 
of conduction in the concrete slab was solved using the same methodology as that of flow. In this work, we 
based on an algorithm that interacts non stationary solid medium with a fluid medium consisting of permanent a 
state by ensuring equal flows and temperatures on the common interface between the two mediums at every 
moment. The numerical simulation of heat transfers and the thermal behavior of the heating slab were analyzed 
for various parameters influencing thermal diffusion. The results obtained show that the numerical 
methodology adopted for the control of fluid-structure coupling is acceptable in comparison with the literature.  

Keywords: heated slab; thermal coupling; fluid flow; heat transfer; not stationary processes 

1. INTRODUCTION 

One of the renewable energy sources is solar energy, which is the most used within cities in 
the Mediterranean. Passive solar is a process of generating thermal energy by converting 
solar radiation into heat. Solar energy is the most developed compared to other renewable 
energies [1–3]. However, the behavior of conversion systems for this energy type is highly 
dependent on variations in climatic parameters such as temperature, solar irradiation and 
storage means [4]. 

In the context of thermal coupling, Giles and al. [5] have studied the numerical stability 
and procedures of fluid-structure thermal coupling. The aim of this study is to analysis the 
thermal diffusion phenomenon with a continuity of temperature and a heat quantity at the 
interface. Birken and al. [6] highlighted the importance of fluid-structure, thermal coupling 
in industrial cooling processes for the steel heat treatment [6]. This numerical study is 
devoted to the thermal interaction study between fluid and structure in the heating slab, also 
called conjugated heat transfer. Monge and Birken [7] have considered two areas with jumps 
in the material conductivity coefficient through the connection interface. Heuzé and al. [8] 

                                                 
* Corresponding author: abdellatif.mebarek@univ-adrar.edu.dz 



100                                            OUDRANE A. et al. 

Copyright ©2023 ASSA                                                                                    Adv. in Systems Science and Appl. (2023) 

have developed a digital tool to simulate the thermomechanical coupling procedure based on 
a fluid-structure coupling to describe the state of the structure matter throughout. 

Currently, numerical simulation of fluid-structure interaction problems is one of the 
biggest challenges for modern scientific computing. Typical examples are found in 
aeronautics, where air flow around an elastic aircraft or air-sheet oscillations in air flow are 
well presented in the work of Dowell and al. [9]. In addition, this interaction is important in 
the in-Turbomachinery field, where the energy transfer will take place between a rotor and 
air [10]. Furthermore, in the field of biomechanics, the elastic behavior of micro-pump or 
artificial membranes in blood flow is affected by this type of interaction by referring to the 
work of Scotti and al. [11] and then Tezduyar and al. [12]. 

Underfloor heating is a technique that provides good comfort while minimizing energy 
consumption. In this context, the aim of this study is to characterize the variable heat 
exchange between a laminar flow of fluid in forced convection and a concrete slab of 
considerable thickness, the top of which is subjected to a constant ambient temperature of 
28°C. 

The modelling is based on the thermal balance calculation at the level for system 
elements: fluid-structure. Model validation was performed using the results obtained by 
Andreo and al [13]. The latter used the same heating system with a heat supply provided by 
solar energy. The second step in this work is to test the parameters influencing the heat 
transfer within the heating slab, analysing the system thermal behavior. 

As part of this work, we propose to exploit this energy potential in the habitat through the 
use of a closed-circuit heating floor. Studies on floor heating technology are numerous in the 
North African region. We are interested in the case of a solar heating system for a single-
zone space in a dry climate similar to that studied by Mokhtari and al [14]. The system is 
equipped with a concrete slab to store and produce heat from the floor within a habitable 
envelope. The principle of operation is to circulate directly into the concrete slab a fluid 
heated by solar collectors [15]. In fact, this slab will have the diffuser role of a soft and 
homogeneous heat throughout the house. 

2. PHYSICAL DESCRIPTION OF THE SLAB STUDIED 

The element of the numerical modelling is a hydronic heating floor which consists of three 
layers with a coil tube (Figure 1). The insulation material is a plate of expanded polystyrene, 
which is the most used in this kind of habitable constructions. The insulation layer height is 5 
cm. Above it, we have a concrete screed 10 cm high, 1m long and 1m wide. In the latter are 
arranged the cross-linked polyethylene tubes [16,17]. They are very often used for the heated 
floors realization. In fact, these semi-rigid pipes are flexible, and they do not need welding to 
be carried out like those of copper. The tubes are arranged in (U) shape with a diameter (d) 
of 20 mm and a 10 cm for spacing [17]. A layer of concrete coating is superimposed on the 
heating grid. 

 
 

Fig.1. Physical description of the slab studied 



                          NUMERICAL INVESTIGATION OF FLUID-STRUCTURE…   101 

Copyright ©2023 ASSA.                                                                                    Adv. in Systems Science and Appl. (2023) 

3. MATHEMATICAL FORMULATION 

3.1. Simplifying hypotheses 

A set of assumptions is retained in this study to simplify the mathematical modelling for 
thermal transfer model. These assumptions are derived from the physical properties of fluid 
flow in a horizontal pipe embedded in a concrete slab. 

The main assumptions taken into account in this study are as follows: 
 The fluid is Newtonian, assumed to be viscous and incompressible; 
 The flow is transient with a laminar regime; 
 Viscous dissipation is negligible: the diffusion of purely mechanical energy is 

neglected because the water speed and viscosity are low; 
 Flow has only two velocity components: one longitudinal speed and the other 

transverse; 
 No internal heat sources 0s   ; 

 The physical properties (µ, Cp, ρ, λ) are constant; 
 Low pipe thickness is neglected in numerical calculations; 
 The fluid-structure interface is in thermodynamic equilibrium; 
 The solid medium to be isotropic. 

 

3.2. Fluid flow modelling in the pipe 

We initially opted for a flow study in a rectangular cross-section duct. This involves the 
flow of viscous fluid between two long plates (L), parallel and separated by a small distance 
(d). Both plates are fixed, and the fluid is moved by a pressure gradient (Figure 2). The 
solution governing a flow of Poiseuille for maximum speed (u0) in the pipe medium is as 
follows [18]: 


















2

0
41)(

d

y
uyu

f
     (1) 

 
Fig.2. Physical description of fluid flow within the pipe 

 
It should be noted that a rigorous treatment of the boundary layer would require the 

complete solution to Navier-Stokes equations. Their complexity prompted Prandtl to 
simplify them to retain only the most important terms. The main idea is to neglect the axial 
gradients (∂/∂x) in front of the transverse gradients (∂/∂y). Thus we obtain the Prandtl 
equations for boundary layer which govern a laminar flow in the heating slab pipe as follows 
[19,20,21]: 

 

 Equation the amount of movement 

0
1

2

2





























y

P
and

y

u
v

x

P

y

u
v

x

u
u

t

u
ff

f

f

f

f

f

f

f

f


   (2) 

 Equation of mass conservation 

0









y

v

x

u
ff       (3) 



102                                            OUDRANE A. et al. 

Copyright ©2023 ASSA                                                                                    Adv. in Systems Science and Appl. (2023) 

 Energy conservation equation 

2

2

y

T

Cy

T
v

x

T
u

t

T
f

fPf

f

f

f

f

f


























    (4) 

3.3. Mathematical model of thermal diffusion 

The floor slab is regarded as a homogeneous solid to which the classical equation of heat 
diffusion is applied [22]. Heat diffusion is defined as a heat transmission mode in a solid 
caused by a temperature difference between two regions of this solid medium [23,24]. The 
numerical modelling is based on the two-dimensional study of heat conduction within the 
heating slab. The equation of thermal conduction is: 

























2

2

2

2

y

T

x

T

Ct

T
bb

bb

bb

P


    (5) 

This equation is discredited by the finite difference method using a three-point forward 
scheme in the contact interface. After the discretization, we get a tridiagonal algebraic 
system, the resolution of which is done by the Thomas algorithm. 

3.4. Initial and boundary conditions 

 At the moment t = 0, the field of calculation under consideration is initialized by: 
0)0(;0)0(;0)0( 

fff
Tvu     (6) 

0)0( 
b

T       (7) 

 The fluid velocity at the channel inlet is given by x = 0 and 0 ≤ y ≤ d; 


















2

0
41)(

d

y
uyu

f
     (8) 

 At the canal outlet we have x = L and 0≤ y ≤ d; 

00;0 














x

T
and

x

v

x

u
fff

     
(9) 

 The coupling interface (fluid-structure) is governed by the following expression 
0<x< L and y=d; 

 

dy
y

T

dy
y

T

f
f

f

b
b

b
b

f










      (10) 

where
fb

and   are respectively the thermal conductivities of the solid and the fluid. 

 At the ambient air-slab interface we have 0< x < L and y = d; 

 airbc

dbyb

b
b TTh

y

T









      (11) 

 

where hc convective coefficient of exchange with ambient air in (W.m-2.k-1), 
b
  is the 

concrete thermal conductivity in (W.m-1.K-1). 
 At the left edge of the slab (adiabatic walls) x = 0 and d ≤ y ≤ d+ep; 

0




dby

b

y

T
      (12) 

 

 At the right border of the slab (adiabatic walls) x = L and d ≤ y ≤ d+ep; 

0




dby

b

y

T
      (13) 

 At the bottom surface of the tube (adiabatic walls) Y = 0 and 0 ≤ x ≤ L; 



                          NUMERICAL INVESTIGATION OF FLUID-STRUCTURE…   103 

Copyright ©2023 ASSA.                                                                                    Adv. in Systems Science and Appl. (2023) 

0




Lfx

f

y

T
      (14) 

  

4. COUPLING METHODOLOGY 

Steady-state fluid-structure coupling has been treated many times. It involves coupling a 
Navier-Stokes solver to a conduction solver and searching for a stationary state in fluid and 
structure medium. It is to highlight that that this type of problem has already been studied 
digitally by Errera and al. [25]. While using the same tools, the work we present here is of a 
different nature. Indeed, we are interested in the non-stationary coupling and more precisely 
in the detailed description of the thermal exchanges in a concrete slab subjected to 
convective flow due to laminar flow. 

4.1. Fluid-Structure Interface 

Figure 3 shows the interface between two mediums with different thermal conductivities. 
The distance between the node (J) and the neighbouring nodes (J-1 and K+1) is calculated 
according to the space pitch [26,27,28]. 

 
Fig. 3. Spatial discretization at the fluid-structure interface. 

 
A finite difference formulation based on the Galerkin method was used to solve the 

equations that govern the fluid-structure interaction model in the heating slab [29]. In perfect 
contact, let us consider the case where the parietal transfer is entirely convective-conductive 
in nature. Regardless of the mode of convection, the transfer of energy between the surface 
of a solid body at temperature (Tb) and the fluid is done by thermal conduction since the fluid 
velocity is zero at the solid body surface. The density continuity the energy flow at the 
surface thus allows writing [30,31,32]: 

 

00

)2.(













fyf

f

f

byb

b

bmW y

T

y

T
     (15) 

 

The discretization of the energy flow, density terms of the lower surface of the solid using 
an implicit method with finite differences to be a three-point forward scheme is given by the 
following expression: 

b

bbb

b

byb

b

b
y

KITKITKIT

y

T












.2

)2,()1,(.4),(.3

0

    (16) 

On the other hand, the density terms discretization for energy flow at in upper surface of 
the fluid using of an implicit method with finite differences with a three-point back pattern is 
given by the following expression: 

f

fff

f

fyf

f

f
y

JITJITJIT

y

T












.2

)2,()1,(4),(.3

0

    (17) 



104                                            OUDRANE A. et al. 

Copyright ©2023 ASSA                                                                                    Adv. in Systems Science and Appl. (2023) 

We pose: 

f

b
a

f
a

a

a

y

y
yand

y
C b















,     (18) 

Or ay  is the relationship between the fluid spatial step and the solid spatial step, a is 

the ratio between the solid thermal conductivity and the fluid thermal conductivity, fT  is the 

fluid temperature and bT  is the solid temperature.  

After an equality and simplification between the two equations (16) and (17), we find the 
expression, of the temperature of fluid-structure coupling for the interface nodes which is 
given as follows: 

)1.(3

))1,(.4)2,(())2,()1,(.4.(
)1,(),(

C

JITJITKITKITC
ITNJIT ffbb

bf 




 
(19) 

5. DIMENSIONAL VARIABLES OF LOCAL EQUATIONS 

To put back the equations governing the flow of the simpler fluid. A set of reduced 
variables (Table 1) were introduced to produce an exploitable non-dimensional form. 

Table.1. Non-dimensional variables of local equations [33]. 
 

Sizes Time Length Velocity Pressure Temperature 
Ladder t  L u  P  T  
Associated 

non-dimensional 
sizes 

t
u

t
d

 0

 dd

y
yet

x
x 

 0
u

u
u 

 
2
0

0

.

).(2

u

PP
P








 0
T

T
T 

 
d The pipe reference diameter m 
u0 The maximum reference velocity at the pipe entrance m.s-1 

t0  = d/u0 Non-dimensional reference time s 
T0 Reference ambient temperature °C 

ρ.u0
2 The reference pressure Pa 

5.1. Non-dimensional equations 

In order to generalize the results, the equations governing for physical phenomenon are 
written in dimensionless forms based on Table 1. Non-dimensional variables are pipe 
diameter and input sizes (velocity and temperature). In the channel, for a two-dimensional 
case in cartesian coordinates, the equations system of heat and mass transfers by forced 
convection are formulated in the following dimensionless form [20]: 

 Quantity equation of the fluid movement; 

2

2 *

.
Re

1
**

*
*

*
*

















y

u

x

P

y

u
v

x

u
u

t

u fff
f

f
f

f    (20) 

 The conservation equation of fluid mass; 

0

**








y

v

x

u ff      (21) 

 The conservation equation of energy in the fluid; 
  

2

2 *

.
Re.Pr

1
*

*
*

*
*














y

T

y

T
v

x

T
u

t

T ff

f

f

f

f

    
(22) 

 Equation of energy conservation in the solid domain (the slab); 



                          NUMERICAL INVESTIGATION OF FLUID-STRUCTURE…   105 

Copyright ©2023 ASSA.                                                                                    Adv. in Systems Science and Appl. (2023) 


































2

2

2

2

*

*

*

*

Re

1
*

*

y

T

x

T

PCt

T bbb

fbb

b




    (23) 

 

   

   

  














































































































2*

2

2*

2

*

2*

2

2*

2

0
*

2*

2

2*

2

0
*

**

.
Re

1
.

Pr

1
.

.

*

**

.
.

.
1

.
.

.
.

*

**

..
.

1
.

.
.

.

*

y

T

x

T

Cpt

T

y

T

x

T

UdCpCpt

T

y

T

x

T

UdCpCpt

T

bb

a

a
b

bb

f

f

ff

f

a

a
b

bb

f

f

ff

f

a

a
b



























   (24) 

 

Or (ρ.Cp)a  is the ratio between the fluid heat capacity and that of the solid and df : height 
of the rectangular pipe. 

5.2. Initial conditions and non-dimensional limits 

In order to write the conditions at the boundary of two mediums in non-dimensional 
forms, we enter the variables in Table 1 in the equations (6 to 14) and we obtain the 
following expressions: 

5.2.1. Non-dimensional initial conditions 

 At the moment (t*), the domain of computation considered is initialized by; 
 

0

*

0

*

0

* )0(;)0(;)0(
T

T
T

u

v
v

u ff

f

f

f u
u      (26) 

0

* )0(
)0(

T

T
T b

b        (27) 

5.2.2. Non-dimensional boundary conditions 

 The fluid velocity at the channel input is given by x* = 0 and 0 ≤ y* ≤ 1; 

















 

2*

*

0

*

1
41)(

y
uy

f
u      (28) 

 At the channel output, we have x * = (L / d) and 0 ≤ y* ≤ 1; 

00
*

*

*

*











x

v
and

x

u
ff

     
(29) 

 

 At the fluid-structure coupling interface is governed by 0 <x*<(L / d) and y* = 0; 

11 *
*

*

*
*

*












f
f

f

b
b

b
a

y
y

T

yy

T
      (30) 

 At the top surface of the slab [27], we have 0 <x* < (L/d) and y* = 1 + (ep/d); 

 **

)/(1

*

1
*

* airb

depy

b
a TT

y

T

b
b









     (31) 

With:  
(h*

c=hc/h0 ) is the convective exchange coefficient with the ambient air. 



106                                            OUDRANE A. et al. 

Copyright ©2023 ASSA                                                                                    Adv. in Systems Science and Appl. (2023) 

 

 At the slab left border (adiabatic walls) x* = 0 and 1 ≤ y ≤ 1+(ep/d); 

0
*

*

*






dby

b

y

T
     (32) 

 At the slab right boundary (adiabatic walls) x* = L and 1≤ y ≤ 1+ (ep/d); 

0
*

*

*






dby

b

y

T
     (33) 

 At the tube bottom surface (adiabatic walls) Y* = 0 and 0 ≤ x* ≤ L/d; 

0
*

*

*






Lfx

f

x

T
     (34) 

6. FLOW CHART OF FLUID-STRUCTURE THERMAL COUPLING 

The flowchart in Figure 4 shows the procedure for calculating the temperature at the slab 
top surface by following the following steps: 

 Requisitions of the physico-thermal properties for fluid-structure system; 
 Initialization of the system at room temperature (28 °C); 
 Resolution of Navier-Stokes equations in the field of fluid; 
 Solving the of the conduction equation in the slab; 
 Coupling temperature test; 
 Test on the calculation overall time; 
 Recording the results obtained. 

7. DATA FROM THE NUMERICAL SIMULATION 

Table 2 shows the physico-thermal properties of air, concrete and water, which are used in 
numerical simulation. 

Table.2. Physico-thermal properties of the studied domain [14,13,34,35]. 
Physico-thermal properties Air Water(ref) Concrete 

Cp  Specific heat in (J.Kg-1.K-1) 1006 4182 880 

  Density in (Kg.m-3) 1,177 1000 2000 

  Dynamic viscosity in (Kg.m-1.s-1) 1,85.10-5 10-3 - 

  Kinematic viscosity in (m2.s-1) 1,75.10-5 10-6 - 

  Thermal conductivity in (W.m-1.K-1) 0,0262 0,597 1,75 

Pr  Number of Prandtl 0,708 7,01 - 
 

The constants used for numerical resolution of the problem studied are [13]; 
 Heat exchange coefficient: hc=8,12 W.m-2 .k-1; 
 Ambient air temperature: Tair=28°C ; 
 Fluid inlet temperature: Te = 60 ° C; 
 Reynolds number: Re = 500; 
 Step time of time: Δt = 10s. 

 



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Fig.4. Flowchart of the numerical modeling of fluid-structure thermal coupling. 

The data of the calculation grid in the two fluid-structure subdomains are presented in 
Table 3: 

Table. 3. Calculation grid data in the fluid-structure domain. 
 Length (m) Height (mm) NI J NK ∆X ∆Y Time step ∆t (s) 

Fluid 
1 

20 101 41 - 
10-2 

5x10-4 
10 

Structure 10 101 - 81 1,25x10-3 
NI Node indices on the x-axis of the two calculation domains. 
NJ Node indices on the y-axis of the fluid domain. 
NK Node indices on the y-axis of the solid domain. 
∆X Step space on the x-axis of the two calculation domains. 
∆Y Step space on the y-axis of the two calculation domains. 
∆t Calculation increment time. 

8. RESULTS AND DISCUSSIONS 

8.1. Validation of the numerical model 

Figure 5 shows the temperature evolution of a point on the slab upper surface as a 
function of time. It is noted that there is an acceptable agreement between the results 
obtained by the developed model and those of Andreo and al [13], so Mokhtari and al [14] 
with relative errors equal to 5,04% and 1,77% respectively. 



108                                            OUDRANE A. et al. 

Copyright ©2023 ASSA                                                                                    Adv. in Systems Science and Appl. (2023) 

 
Fig.5. Evolution of the temperature profile at a point in the slab medium as a function of the overall time. 

8.2. Evolution of the concrete slab temperature according to the spacing step 

Figure 6 shows the variation in the heating slab temperature as a function of the space 
pitch ratio for four times. This variation has been highlighted for three different positions in 
the system (at the inlet, the medium and at the outlet). We can notice that: 

 The evolution of thermal diffusion in the concrete slab is almost similar for the three 
positions considered (inlet, medium and outlet); 

 There is a remarkable temperature difference between the positions in the system for 
the four selected global calculation times;  

 For a ratio of space varying between 0 and 0.12 with a fluid inlet temperature in the 
pipe at 60°C, the heat propagation in the solid decreases almost to temperature values 
(in the medium) of about 38°C for t = 3600s (Fig. 6.a), 47°C for t = 10800s (Fig. 6.b), 
48°C for t = 14400s (Fig. 6.c) and 49°C for t = 43200s (Fig. 6.d). 
 

   
 



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Fig.6. Comparison between the three profiles (input, medium and output) of the temperature as a function 

of the slab space pitch ratio for (a) t = 3600s, (b) t = 10800s, (c) t = 14400s and (d) t = 43200s 

8.3. Evolution of the system temperature for different thicknesses of the slab 

Figure 7 illustrates the change in temperature in the medium slab as a function of time for 
different thicknesses of the concrete top layer. It is clear that the heat propagation in the slab 
decreases considerably with the increase in the thickness of the upper layer the concrete. 
That is to say, when the concrete thickness is decreased (for example 10 cm), the evolution 
of the temperature increases rapidly as a function of time and vice versa. In other words, the 
heat diffusion in the room will be even lower than the concrete mass is important. 

 
Fig.7. Evolution of the temperature profile in the slab as a function of time for different thicknesses of the 

concrete 

8.4. Effect of variation of the convective exchange coefficient with the ambient air 

Figure 8 shows the evolution of heating slab temperature as a function of time for 
different convective coefficients with ambient air. It can be observed that when the value of 
the convective exchange coefficient increases, the thermal diffusion increases slightly and 
vice versa. It should also be noted that the variation for convective exchange coefficient with 
ambient air has a notable influence on the thermal diffusion of heat within the heating slab. 
In addition, we will note that the evolution of the temperature will start from 28°C, the 
atmosphere temperature , and it will increase over time to the highest value around 60°C, 
which is the temperature at which the fluid inlet the slab pipe. 

 



110                                            OUDRANE A. et al. 

Copyright ©2023 ASSA                                                                                    Adv. in Systems Science and Appl. (2023) 

 
Fig.8. Evolution of the heating slab temperature as a function of time for different convective coefficients 

with ambient air 

8.5. Evolution of the velocity profile of the fluid in the pipeline 

The evolution of fluid velocity within the pipe as a function of the space step for different 
Reynolds numbers is illustrated in Figure 9. In this figure, it can be seen that the shape of the 
fluid velocity profile in the pipe is parabolic. In addition, it keeps the same shape, but its 
maximum value increases with the increase in the number of Reynolds or it reaches a 
maximum value equal to 0,0175m/s for Re = 800. This evolution shows that the variation 
influence of the Reynolds number on the fluid flow is important. 
 

 
Fig. 9. Profiles of fluid velocities within the pipe versus space step for different Reynolds numbers 

9. CONCLUSION 

The objective of this digital study is the adaptation of a numerical methodology in order 
to be able to control and characterize the thermal exchanges within a heating slab intended 
for heating by the ground. In fact, we analyzed the temperature evolution at the inlet, 
medium and outlet of the system according to the space step. In addition, we analyzed the 
evolution of the slab temperature for different thicknesses of the concrete and also the effect 
of variation of the convective exchange coefficient with the ambient air on the surface 
temperature of the slab. On the other hand, we have highlighted the effect of the Reynolds 
number variation on the evolution of the fluid velocity profile in the pipe. 

The validity of this numerical approach is confirmed based on the work done by Mokhtari 
and Andreo. The methodology we have presented makes it possible to analyses the thermal 



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transient phase over very short periods of time using the current numerical model of fluid-
structure thermal coupling. 
 

Nomenclature 
Cp  Heat capacity. 11 ..  KmW  
d  Pipe diameter. m  
Ep  Concrete thickness. m  

hc  Convective coefficient with ambient air. 12 ..  KmW  

0
h  Convective coefficient reference. 12 ..  KmW  

I  Index of the abscissa axis incrementation. - 

L  Pipe length. m  

J  Index of the ordinate axis incrementation in the fluid domain. - 
k  Increment index of the y-axis in the solid domain. - 

Pr Number of Prandtl. - 

f
P  Fluid pressure. Pas  

s
  The internal heat quantity. 2. mW  

Re Number of Reynolds. - 

air
T  Air temperature. K  

e
T  Inlet temperature of the fluid in the pipe. K  

milieu
T  Temperature in the slab middle. K  

entrée
T  Temperature at the pipe entrance. K  

sortie
T

 
Temperature at the pipe outlet. K  

*T  Dimensional temperature. - 

f
T  Fluid temperature. K  

b
T  Concrete temperature. K  

t  Time. s  

0
U          Initial longitudinal velocity of the fluid. 1. sm  

*U  Dimensional longitudinal velocity of the fluid. - 

f
V  Initial transverse velocity of the fluid. 1. sm  

*V  Initial transverse velocity of the fluid. - 
y Directional normal of the fluid diffusivity. - 
Greek symbols 
  Thermal conductivity. 11 ..  KmW  

a  Report between of the fluid thermal conductivity and the solid. - 

  Dynamic viscosity of the fluid. 
11..
 SmKg  

  Volumic mass. 3. mKg  
 

REFERENCES 

[1] Mehdary A.,  Naamane A., M'sirdi K. (2011), ‘’Real Time Implementation Control for 
multi-source energy system’’, Sustainability in Energy and Building, Page 524. 

[2] Ky Le. (2008),’’ Gestion optimale des consommations d'énergie dans les bâtiments’’. 
Thesis P.h.D, soutenu le 10 Juillet 2008. Laboratoire de génie électrique, Institut 
Polytechnique de Grenoble, la France. 

[3] Tina G. and Scrofani S. (2008), ‘’ Electrical and thermal model for PV module 
temperature evaluation’’. IEEE Electro technical Conference, Pages 585-590. 

[4] Abarkan M. (2014), ‘’Modélisation et Analyse du comportement d'un Bâtiment équipé 
d'un Système Multi Sources d'énergie’’, Thesis P.h.D de l’Université Aix-Marseille, la 
France. 



112                                            OUDRANE A. et al. 

Copyright ©2023 ASSA                                                                                    Adv. in Systems Science and Appl. (2023) 

[5] Giles M. B. (1997), ‘’Stability analysis of numerical interface conditions in fluid-
structure thermal analysis’’, International journal for numerical methods in fluids, vol. 
25, 421-436. 

[6] Birken P., Quint K., Hartmann S. and Meister A. (2010), A time-adaptive fluid-structure 
interaction method for thermal coupling’’, Comput Visual Sci.(2010) 13:331–340, DOI 
10.1007/s00791-010-0150-4. 

[7] Monge A. and Birken P. (2018), ‘’On the convergence rate of the Dirichlet–Neumann 
iteration for unsteady thermal fluid–structure interaction’’, Comput Mech (2018) 62- 
525, doi.org/10.1007/s00466-017-1511-3. 

[8] Heuzé T., Leblond J. B. and. Bergheau J. M. (2011), ‘’Modélisation des couplages 
fluide/solide dans les procédés d’assemblage à haute température’’, Mécanique & 
Industries (12), 183–191. 

[9] Dowell E., Thomas J. and Hall K. (2004), ‘’Transonic limit cycle oscillation analysis 
using reduced order aerodynamic models’’, Journal of Fluids and Structures (19), 17–
27. 

[10] Willcox K., Paduano J. and Peraire J. (1999), ‘’Low order aerodynamic models for 
aeroelastic control of turbomachines’’. In 40th AIAA ASME ASCE AHS ASC structures, 
structural dynamics and materials conference, St Louis, MO. 

[11] Scotti C. and Finol E. (2007), ‘’Compliant biomechanics of abdominal aortic 
aneurysms: a fluid–structure interaction study’’, Computers and Structures, (85), 1097–
1130. 

[12] Tezduyar T. E., Schwaab M. and Sathe S. (2008), ‘’Sequentially-coupled arterial fluid-
structure interaction (scafsi) technique’’, Computer Methods in Applied Mechanics and 
Engineering. 

[13] Ugo A., Fabien B. and Adeline V. (2015), ‘’Etude de la diffusion de la chaleur dans un 
plancher chauffant en environnement terrestre’’, étude de l’université de Paul Sabatier, 
Toulouse III, la France. 

[14] Mokhtari F., Ait Messaoudène N., Hamid A. and Belhamel M. (2006), ‘’Etude du 
comportement thermique d’une maison munie d’un système de chauffage solaire’’, 
Revue des Energies Renouvelables Vol. 9 N°4, 363 – 370. 

[15] Sobhy I., Brakez A. and Benhamou B. (2014), ‘’Modélisation dynamique du 
comportement thermique d’un plancher chauffant avec circuit fermé à Marrakech’’, 1ere 
Colloque Franco-Marocain sur les Energies Renouvelables, COFMER’01, Rabat 28-30. 

[16] Fiche Technique Tube PER  Prégainé (2016), 91, Rue Duruisseau, PA des Chesnes – 
38297 St Quentin – Fallavier, Date de mise à jour : 22/06/16. 

[17] Oudrane A., Aour B. and Benhamou M. (2016), ‘’Étude et analyse paramétrique d’une 
installation solaire : plancher solaire dicte d'implanter dans la région d’Adrar’’, ElWahat 
pour les Recherches et les Etudes Vol.9 n°1, 27 – 49. 

[18] Noel E. and Delaune M. (2012), ‘’Simulation numérique des équations de Navier-
Stokes’’, Projet de Physique P6-3 STPI/P6-3/2011–39, institut national des sciences 
appliquées de Rouen département sciences et techniques pour l’Ingénieur, de l'université 
-76801 Saint Etienne du Rouvray. 

[19] Marty P. (2012), ‘’Transferts thermiques convectifs’’, Génie des Procédés Master 2, 
Université Joseph Fourier, Grenoble, version modifiée. 

[20] Hassine N. B., Chesneau X. and Laatar A. H. (2017), ‘’Numerical simulation of heat 
and mass transfers during solar drying of sewage sludge: Radiation effect’’, Energy 
Procedia (139), 804–809. 

[21] Abdul Aziz M. S. and Abdullah M. Z. (2014), ‘’Thermal Fluid-Structure Interaction in 
the Effects of Pin-Through-Hole Diameter during Wave Soldering’’, Hindawi 
Publishing Corporation Advances in Mechanical Engineering Volume 2014,Article ID 
275735, 13 pages, doi.org/10.1155/2014/275735. 



                          NUMERICAL INVESTIGATION OF FLUID-STRUCTURE…   113 

Copyright ©2023 ASSA.                                                                                    Adv. in Systems Science and Appl. (2023) 

[22] Oudrane A., Aour B., Hamouda M. and Benhamou M. (2016), ‘’Méthodologie pour la 
détermination de l’écartement optimal de la chaîne tubulaire d’une dalle chauffante’’, 
Revue des Energies Renouvelables Vol. 19 N°1, 11 – 19. 

[23] Debard Y. (2011), ‘’Méthode des ´éléments finis : thermique, Master Modélisation 
Numérique et Réalité Virtuelle’’, Université du Mans, la France. 

[24] Chiasson A. D., Spitler J. D. and Rees S. J. (2000), ’’A Model for simulating the 
performance of a pavement heating system as a supplemental heat rejecter with closed-
loop ground-source heat pump systems’’, Journal of Solar Energy Engineering, Vol. 
122,183-191. 

[25] Errera M. P., Chaineray G. and Chemin S. (2007), ‘’Etude du transitoire thermique dans 
un matériau via un couplage convection-conduction’’, Congrès Français de Thermique, 
SFT 2007, Île des Embiez, 29 mai - 1 juin 2007. 

[26] Saatdjian E. (1998), ‘’Phénomènes de transport et leurs résolutions numériques’’, 2éme 
édition, Polytechnica, 15 rue Lacépède F-75005, ISBN2-84045-057-6, Paris, la France.  

[27] Florides G. and Kalogirou S. (2007), ‘’Ground heat exchangers—A review of systems: 
models and applications’’, Renewable Energy (32), 2461–2478. 

[28] Ghalambaz M., Jamesahar E., Ismael A. and Chamkha J. (2017), ‘’Fluid-structure 
interaction study of natural convection heat transfer over a flexible oscillating fin in a 
square cavity’’, International Journal of Thermal Sciences (111), 256-273. 

[29] Al-Amiri A. and Khanafer K. (2011), ‘’Fluid–structure interaction analysis of mixed 
convection heat transfer in a lid-driven cavity with a flexible bottom wall’’, 
International Journal of Heat and Mass Transfer 54, pp.3826–3836. 

[30] Mergui S. (2010), ‘’Transferts thermiques’’, licence de mécanique 2éme année, UPMC, 
université de Sorbonne. 

[31] Abide S., Viazzo S. and Solliec C. (2018), ‘’Simulation numérique du refroidissement 
d’une plaque plane par un jet plan impactant’’, LAMPS-GME, Université de Perpignan, 
52 Avenue Paul Alduy, 66860 Perpignan, la France. https://docplayer.fr/25679830-
Simulation-numerique-du-refroidissement-d-une-plaque-plane-par-un-jet-plan 
impactant.html, (Page consulté le 29/05/2018). 

[32] Gobin D., Levesque D. and Benard C. (1979), ‘’Stockage de l’énergie solaire : 
simulation numérique du transfert d’énergie par conduction et rayonnement dans un 
milieu à deux phases’’, revue de physique appliquée, tome 14, page 125. 

[33] Mahdaoui H., Chesneau X. and Laatar A. H. (2017), ‘’Numerical simulation of flow 
through a porous cylinder’’, Energy Procedia (139), 785-790. 

[34] Chahwane L. (2011), ‘’Valorisation de l’inertie thermique pour la performance 
énergétique des bâtiments’’, P.h.D Thesis de l’université de Grenoble, la France.  

[35] Oudrane A. (2018), ‘’Contribution à la modélisation et au développement des systèmes 
de chauffage solaire à usage individuel’’, P.h.D Thesis d’Ecole Nationale Polytechnique 
d’Oran - Maurice Audin, Algérie.