Ultrasonic transducers for flow velocity measurement with meta slab
An ultrasonic transducer employing a meta slab includes a piezoelectric body configured to generate elastic waves; a meta slab connected to the piezoelectric body and configured to induce elastic wave mode conversion resonance with respect to the elastic waves incident on the meta slab; and a wedge connected to the meta slab, attached to an external surface of a pipe, and configured to transmit the elastic waves having passed through the meta slab to the pipe. The meta slab includes an anisotropic medium and a thickness of the meta slab satisfies the equation as follows: d=m·n FS ·λ FS /4, d=m·n SS ·λ SS /4, n SS /2−n FS /2=odd. Thus, highly-efficient flow velocity measurement is possible.
1. A flow velocity measuring apparatus including an ultrasonic transducer employing a meta slab, the flow velocity measuring apparatus comprising:
a first piezoelectric body configured to generate elastic waves;
a first meta slab connected to the first piezoelectric body and configured to induce first elastic wave mode conversion resonance with respect to the elastic waves incident on the first meta slab;
a first wedge connected to the first meta slab, attached to an external surface of a pipe, and configured to transmit the elastic waves having passed through the first meta slab to the pipe at a first incidence angle;
a second wedge attached to the external surface of the pipe and configured to receive the elastic waves having passed through the pipe;
a second meta slab connected to the second wedge and configured to induce second elastic wave mode conversion resonance with respect to the elastic waves incident on the second meta slab; and
a second piezoelectric body connected to the second meta slab and configured to receive elastic waves,
wherein
the first meta slab and the second meta slab include anisotropic media and a length of each of the first and second meta slabs satisfies the following:
d
=
m
·
n
FS
λ
FS
4
,
d
=
m
·
n
SS
λ
SS
4
,
n
SS
2
-
n
FS
2
=
odd
(d: length of a unit cell of a microstructure of a meta slab, m: an integer being an odd number, n FS and n SS : coprime integers corresponding to a fast skew mode and a slow skew mode, respectively, λ FS : a fast skew mode wavelength, λ ss : a slow skew mode wavelength, and odd: an integer being an odd number), and
microstructures of the first meta slab and the second meta slab include monolayer or multilayer structures.
2. The flow velocity measuring apparatus of claim 1 , wherein the first meta slab satisfies the following equation:
Z =ρ√{square root over ( C 11 C 66 −C 16 2 )}=√{square root over (ρ 0 A c 11 A ρ 0 B c 66 B )}
(Z: impedance of the first meta slab, ρ: a mass density of the first meta slab, C 11 : an effective longitudinal elastic modulus of the first meta slab, C 66 : an effective shear elastic modulus of the first meta slab, C 16 : a mode-coupling elastic modulus of the first meta slab, ρ 0 A : a mass density of a background medium that elastic waves incident on the first meta slab from the background medium, c 11 A : a longitudinal elastic modulus of the background medium that elastic waves incident on the first meta slab from the background medium, ρ 0 B : a mass density of a background medium that elastic waves enter into the background medium after passing through the first meta slab, and c 66 B : a shear elastic modulus of the background medium that elastic waves enter into the background medium after passing through the first meta slab), and
the second meta slab satisfies the following equation:
Z =ρ√{square root over ( C 11 C 66 −C 16 2 )}=√{square root over (ρ 0 A c 66 A ρ 0 B c 11 B )}
(Z: impedance of the second meta slab, ρ: a mass density of the second meta slab, C 11 : an effective longitudinal elastic modulus of the second meta slab, C 66 : an effective shear elastic modulus of the second meta slab, C 16 : a mode-coupling elastic modulus of the second meta slab, ρ 0 A : a mass density of a background medium that elastic waves incident on the second meta slab from the background medium, c 66 A : a shear elastic modulus of the background medium that elastic waves incident on the second meta slab from the background medium, ρ 0 B : a mass density of a background medium that elastic waves enter into the background medium after passing through the second meta slab, and c 11 B : a longitudinal elastic modulus of the background medium that elastic waves enter into the background medium after passing through the second meta slab).
3. The flow velocity measuring apparatus of claim 2 , wherein
the respective effective longitudinal elastic moduli of the first meta slab and the second meta slab are same as the respective effective shear elastic moduli of the first meta slab and the second meta slab, and
respective wave vibration directions of the fast skew mode and the slow skew mode form +45 degrees and −45 degrees, respectively.
4. The flow velocity measuring apparatus of claim 1 , wherein
the respective effective longitudinal elastic moduli of the first meta slab and the second meta slab are same as the respective effective shear elastic moduli of the first meta slab and the second meta slab, and
respective wave vibration directions of the fast skew mode and the slow skew mode form +45 degrees and −45 degrees, respectively.
5. The flow velocity measuring apparatus of claim 1 , wherein the mode-coupling elastic modulus of each of the first meta slab and the second meta slab satisfies the following equation:
C 16 ≠0
C 16 <( C 11 +C 66 )/4
(C 11 : an effective longitudinal elastic modulus of a meta slab, C 66 : an effective shear elastic modulus of the meta slab, and C 16 : a mode-coupling elastic modulus of the meta slab).
6. The flow velocity measuring apparatus of claim 5 , wherein
the first meta slab which causes complete mode conversion satisfies the following four equations
d
=
m
·
n
FS
λ
FS
4
,
d
=
m
·
n
SS
λ
SS
4
,
n
SS
2
-
n
FS
2
=
odd
Z
=
ρ
C
11
C
66
-
C
16
2
=
ρ
0
A
c
11
A
ρ
0
B
c
66
B
Z
=
ρ
C
11
C
66
-
C
16
2
=
ρ
0
A
c
66
A
ρ
0
B
c
11
B
C
16
≠
0
C
16
<
(
C
11
+
C
66
)
/
4
and the effective longitudinal elastic modulus is identical with the effective shear elastic modulus, and satisfies the following equation:
C
1
1
=
Z
0
ρ
η
1
+
η
(
n
SS
n
F
S
+
n
F
S
n
S
S
)
C
6
6
=
Z
0
ρ
η
1
+
η
(
n
SS
n
F
S
+
n
F
S
n
SS
)
C
1
6
=
Z
0
ρ
η
(
1
+
η
)
2
(
n
SS
n
F
S
+
n
F
S
n
SS
)
2
-
1
ρ
=
Z
0
n
F
S
n
SS
4
f
F
M
C
d
Z
0
=
ρ
0
A
c
1
1
A
ρ
0
B
c
6
6
B
η
=
C
1
1
C
6
6
(C 11 : an effective longitudinal elastic modulus of the first meta slab, C 66 : an effective shear elastic modulus of the first meta slab, C 16 : a mode-coupling elastic modulus of the first meta slab, ρ: a mass density of the first meta slab, n FS and n SS : coprime integers corresponding to a fast skew mode and a slow skew mode, respectively, ρ 0 A : a mass density of a background medium that elastic waves incident on the first meta slab from the background medium, c 11 A : a longitudinal elastic modulus of the background medium that elastic waves incident on the first meta slab from the background medium, ρ 0 B : a mass density of a background medium that elastic waves enter into the background medium after passing through the first meta slab, c 66 B : a shear elastic modulus of the background medium that elastic waves enter into the background medium after passing through the first meta slab, f FMC : a set frequency at which complete mode conversion occurs, and d: a unit cell length of a microstructure of a meta slab), and
the second meta slab which causes complete mode conversion satisfies the following four equations
d
=
m
·
n
FS
λ
FS
4
,
d
=
m
·
n
SS
λ
SS
4
,
n
SS
2
-
n
FS
2
=
odd
Z
=
ρ
C
11
C
66
-
C
16
2
=
ρ
0
A
c
11
A
ρ
0
B
c
66
B
Z
=
ρ
C
11
C
66
-
C
16
2
=
ρ
0
A
c
66
A
ρ
0
B
c
11
B
C
16
≠
0
C
16
<
(
C
11
+
C
66
)
/
4
and the effective longitudinal elastic modulus is identical with the effective shear elastic modulus, and satisfies the following equation:
C
1
1
=
Z
0
ρ
η
1
+
η
(
n
SS
n
F
S
+
n
F
S
n
S
S
)
C
6
6
=
Z
0
ρ
η
1
+
η
(
n
SS
n
F
S
+
n
F
S
n
SS
)
C
1
6
=
Z
0
ρ
η
(
1
+
η
)
2
(
n
SS
n
F
S
+
n
F
S
n
SS
)
2
-
1
ρ
=
Z
0
n
F
S
n
SS
4
f
F
M
C
d
Z
0
=
ρ
0
A
c
66
A
ρ
0
B
c
11
B
η
=
C
1
1
C
6
6
(C 11 : an effective longitudinal elastic modulus of the second meta slab, C 66 : an effective shear elastic modulus of the second meta slab, C 16 : a mode-coupling elastic modulus of the second meta slab, ρ: a mass density of the second meta slab, n FS and n SS : coprime integers corresponding to a fast skew mode and a slow skew mode, respectively, ρ 0 A : a mass density of a background medium that elastic waves incident on the second meta slab from the background medium, c 66 A : a shear elastic modulus of the background medium that elastic waves incident on the second meta slab from the background medium, ρ 0 B : a mass density of a background medium that elastic waves enter into the background medium after passing through the second meta slab, c 11 B : a longitudinal elastic modulus of the background medium that elastic waves enter into the background medium after passing through the second meta slab, f FMC : a set frequency at which complete mode conversion occurs, and d: a unit cell length of a microstructure of a meta slab).
7. The flow velocity measuring apparatus of claim 5 , wherein,
when the material of the pipe is polymer resin selected from a group of polymer resins, respective materials of the first and second wedges are also polymer resin, and
when the material of the pipe is metal, the respective materials of the first and second wedges are also aluminum or stainless steel.
8. A flow velocity measuring method including an ultrasonic transducer employing a meta slab, and the flow velocity measuring method comprising:
generating elastic waves, wherein the generating is performed by a first piezoelectric body;
receiving the elastic waves generated by the first piezoelectric body, wherein the receiving is performed by a first meta slab that includes an anisotropic medium and has a length satisfying the following equation:
d
=
m
·
n
FS
λ
FS
4
,
d
=
m
·
n
SS
λ
SS
4
,
n
SS
2
-
n
FS
2
=
odd
(d: a length of a unit cell of a meta slab, m: an integer being an odd number, n FS and n SS : coprime integers corresponding to a fast skew mode and a slow skew mode, respectively, λ FS : a fast skew mode wavelength, λ SS : a slow skew mode wavelength, and odd: an integer being an odd number);
receiving the elastic waves having passed through the first meta slab, wherein the receiving is performed by a first wedge;
making the elastic waves be incident on a pipe at a first incidence angle, wherein the incidence of the elastic waves is performed by the first wedge;
receiving the elastic waves having passed through the pipe, wherein the receiving is performed by a second wedge;
receiving the elastic waves having passed through the second wedge, wherein the receiving is performed by a second meta slab that includes an anisotropic medium and has a length satisfying the preceding equation:
sensing the elastic waves having passed through the second meta slab, wherein the sensing is performed by a second piezoelectric body; and
determining a flow velocity of the fluid flowing in the pipe according to the following equation by using the sensed elastic waves:
V
=
D
2
sin
θ
3
L
cos
θ
3
L
(
1
Δ
t
1
-
1
Δ
t
2
)
(D: an inner diameter of the pipe, θ 3 L : a longitudinal wave transmission angle in water, Δt 1 : a signal time difference when a direction of elastic waves is the same as a flow direction of the fluid, and Δt 2 : a signal time difference when the direction of the elastic waves is opposite to the flow direction of the fluid).
9. The flow velocity measuring method of claim 8 , wherein,
in the receiving of the elastic waves generated by the first piezoelectric body and the receiving of the elastic waves having passed through the first meta slab, the first piezoelectric body, the first meta slab, and the first wedge satisfy the following equation, and thus impedance matching is performed,
Z =ρ√{square root over ( C 11 C 66 −C 16 2 )}=√{square root over (ρ 0 A c 11 A ρ 0 B c 66 B )}
(Z: impedance of the first meta slab, ρ: a mass density of the first meta slab, C 11 : an effective longitudinal elastic modulus of the first meta slab, C 66 : an effective shear elastic modulus of the first meta slab, C 16 : a mode-coupling elastic modulus of the first meta slab, ρ 0 A : a mass density of a background medium that elastic waves incident on the first meta slab from the background medium, c 11 A : a longitudinal elastic modulus of the background medium that elastic waves incident on the first meta slab from the background medium, ρ 0 B : a mass density of a background medium that elastic waves enter into the background medium after passing through the first meta slab, and c 66 B : a shear elastic modulus of the background medium that elastic waves enter into the background medium after passing through the first meta slab); and
in the receiving of the elastic waves having passed through the pipe and the receiving of the elastic waves having passed through the second wedge, the second wedge, the second meta slab, and the second piezoelectric body satisfy the following equation below, and thus impedance matching is performed,
Z =ρ√{square root over ( C 11 C 66 −C 16 2 )}=√{square root over (ρ 0 A c 66 A ρ 0 B c 11 B )}
(Z: impedance of the second meta slab, ρ: a mass density of the second meta slab, C 11 : a effective longitudinal elastic modulus of the second meta slab, C 66 : a effective shear elastic modulus of the second meta slab, C 16 : a mode-coupling elastic modulus of the second meta slab, ρ 0 A : a mass density of a background medium that elastic waves incident on the second meta slab from the background medium, c 66 A : a shear elastic modulus of the background medium that elastic waves incident on the second meta slab from the background medium, ρ 0 B : a mass density of a background medium that elastic waves enter into the background medium after passing through the second meta slab, and c 11 B : a longitudinal elastic modulus of the background medium that elastic waves enter into the background medium after passing through the second meta slab).
10. The flow velocity measuring method of claim 9 , wherein
the respective effective longitudinal elastic moduli of the first meta slab and the second meta slab are same as the respective effective shear elastic moduli of the first meta slab and the second meta slab, and
respective wave vibration directions of the fast skew mode and the slow skew mode form +45 degrees and −45 degrees, respectively.
11. The flow velocity measuring method of claim 8 , wherein
the respective effective longitudinal elastic moduli of the first meta slab and the second meta slab are same as the respective effective shear elastic moduli of the first meta slab and the second meta slab, and
respective wave vibration directions of the fast skew mode and the slow skew mode form +45 degrees and −45 degrees, respectively.
12. The flow velocity measuring method of claim 8 , wherein
the mode-coupling elastic modulus of each of the first meta slab and the second meta slab satisfies the following equation:
C 16 ≠0
C 16 <( C 11 +C 66 )/4
13. An ultrasonic transducer employing a meta slab, the ultrasonic transducer comprising:
a piezoelectric body configured to generate elastic waves;
a meta slab connected to the piezoelectric body and configured to induce elastic wave mode conversion resonance with respect to the elastic waves incident on the meta slab; and
a wedge connected to the meta slab, attached to an external surface of the pipe, and configured to transmit the elastic waves having passed through the meta slab to the pipe at a first incidence angle,
wherein
the meta slab includes an anisotropic medium and the length of the meta slab satisfies the following equation:
d
=
m
·
n
FS
λ
FS
4
,
d
=
m
·
n
SS
λ
SS
4
,
n
SS
2
-
n
FS
2
=
odd
(d: a unit cell length of a microstructure of a meta slab, m: an integer being an odd number, n FS and n SS : coprime integers corresponding to a fast skew mode and a slow skew mode, respectively, λ FS : a fast skew mode wavelength, λ SS : a slow skew mode wavelength, and odd: an integer being an odd number), and
a microstructure of the meta slab includes a monolayer or multilayer structure.
14. The ultrasonic transducer of claim 13 , wherein the meta slab satisfies the following equation:
Z =ρ√{square root over ( C 11 C 66 −C 16 2 )}=√{square root over (ρ 0 A c 11 A ρ 0 B c 66 B )}
(Z: impedance of the meta slab, ρ: a mass density of the meta slab, C 11 : an effective longitudinal elastic modulus coefficient of the meta slab, C 66 : an effective shear elastic modulus of the meta slab, C 16 : a mode-coupling elastic modulus of the meta slab, ρ 0 A : a mass density of a background medium that elastic waves incident on the meta slab from the background medium, c 11 A : a longitudinal elastic modulus of the background medium that elastic waves incident on the meta slab from the background medium, ρ 0 B : a mass density of a background medium that elastic waves enter into the background medium after passing through the meta slab, and c 66 B : a shear elastic modulus of the background medium that elastic waves enter into the background medium after passing through the meta slab).
15. The ultrasonic transducer of claim 14 , wherein
the effective longitudinal elastic modulus and the effective shear elastic modulus of the meta slab are same as each other, and
respective wave vibration directions of the fast skew mode and the slow skew mode form +45 degrees and −45 degrees, respectively.
16. The ultrasonic transducer of claim 13 , wherein
the effective longitudinal elastic modulus and the effective shear elastic modulus of the meta slab are same as each other, and
respective wave vibration directions of the fast skew mode and the slow skew mode form +45 degrees and −45 degrees, respectively.
17. The ultrasonic transducer of claim 13 , wherein a mode-coupling elastic modulus of the meta slab satisfies the following equation:
C 16 ≠0
C 16 <( C 11 +C 66 )/4
18. The ultrasonic transducer of claim 17 , wherein the meta slab which causes complete mode conversion satisfies the following three equations
d
=
m
·
n
FS
λ
FS
4
,
d
=
m
·
n
SS
λ
SS
4
,
n
SS
2
-
n
FS
2
=
odd
Z
=
ρ
C
11
C
66
-
C
16
2
=
ρ
0
A
c
11
A
ρ
0
B
c
66
B
Z
=
ρ
C
11
C
66
-
C
16
2
=
ρ
0
A
c
66
A
ρ
0
B
c
11
B
C
16
≠
0
C
16
<
(
C
11
+
C
66
)
/
4
and the effective longitudinal elastic modulus of the meta slab is identical with the effective shear elastic modulus of the meta slab, and satisfies the following equation:
C
1
1
=
Z
0
ρ
η
1
+
η
(
n
SS
n
F
S
+
n
F
S
n
S
S
)
C
6
6
=
Z
0
ρ
η
1
+
η
(
n
SS
n
F
S
+
n
F
S
n
SS
)
C
1
6
=
Z
0
ρ
η
(
1
+
η
)
2
(
n
SS
n
F
S
+
n
F
S
n
SS
)
2
-
1
ρ
=
Z
0
n
F
S
n
SS
4
f
F
M
C
d
Z
0
=
ρ
0
A
c
11
A
ρ
0
B
c
66
B
η
=
C
1
1
C
6
6
(C 11 : an effective longitudinal elastic modulus of the meta slab, C 66 : an effective shear elastic modulus of the meta slab, C 16 : a mode-coupling elastic modulus of the meta slab, ρ: a mass density of the meta slab, n FS and n SS : coprime integers corresponding to a fast skew mode and a slow skew mode, respectively, ρ 0 A : a mass density of a background medium that elastic waves incident on the meta slab from the background medium, c 11 A : a longitudinal elastic modulus of the background medium that elastic waves incident on the meta slab from the background medium, ρ 0 B : a mass density of a background medium that elastic waves enter into the background medium after passing through the meta slab, c 66 B : a shear elastic modulus of the background medium that elastic waves enter into the background medium after passing through the meta slab, f FMC : a set frequency at which complete mode transformation occurs, and d: a unit cell length of a microstructure of the meta slab).
19. The ultrasonic transducer of claim 17 , wherein,
when the material of the pipe is polymer resin selected from a group of polymer resins, the materials of the wedges are also selected from polymer resin, and
when the material of the pipe is metal, the materials of the wedges are also selected from aluminum or stainless steel.